A few years ago, following the release of her widely acclaimed 2016 album, My Woman, Angel Olsen started noticing celebrities turning up at her live shows.

Christina Hendricks, of Mad Men fame, came to a concert in Los Angeles and swayed along to Olsen’s smouldering, deeply felt folk-rock. She stuck around to say hello afterwards, as did the actor Brie Larson, who proved herself to be, in Olsen’s assessment, “super down-to-earth and really normal and just chill”.

Miley Cyrus, meanwhile, called up ahead of one LA gig and was duly granted a parking space, her own green room and a prime viewing position at the side of the stage. “It was really sweet,” Olsen recalls. “She took selfies with everybody, with my record in her mouth. I was like: ‘This is so stupid!’ It was a fun time.”

You’d think the Missouri-born singer-songwriter, who had built up a passionate following but was still at a remove from the mainstream, would have been gratified by all this starry attention. But Olsen, who has a bone-dry sense of humour, which she expresses in perfect deadpan, recalls the surge of interest that followed My Woman with a raised eyebrow. “There were a lot of people coming out of the woodwork going: ‘I loved the My Woman record.’ I’m like: ‘Yeah…’” She affects to look supremely unmoved. “‘I’ve made records before.’”

By then, Olsen had been putting out music for the best part of a decade. She released her first solo EP in 2010 and followed it with an album of raw, acoustic folk songs entitled Half Way Home. On 2014’s Burn Your Fire for No Witness, she shifted up a gear, embracing the buzz of electric guitars and giving full vent to her extraordinary, multifaceted voice. The NME gave the record 9/10 and called it “sublime”, while Pitchfork, rounding up the best albums of the past decade in December, ranked it at No 26.

But it was My Woman, her third full-length, that brought Olsen to a wider audience, breaking the Billboard top 50 and landing her high-profile festival slots at Glastonbury and Primavera. In an era when pop careers go stratospheric on the strength of a single track, such a gradual rise seems positively old-fashioned – but that’s only fitting for an artist who takes much of her musical inspiration, along with a penchant for beehive hairstyles and sequinned dresses, from the middle decades of the 20th century.

Last October, Olsen released All Mirrors, her biggest, boldest album to date. A brooding, synth-laden opus that harnesses the full firepower of a 12-piece string section, it has been universally acclaimed – for Olsen’s risk-taking as much as the shimmering beauty of her arrangements. “It’s an album that keeps taking ostensibly recognisable musical forms and twisting them out of shape into something challenging and intriguing,” wrote the Guardian’s Alexis Petridis.

We meet in Lisbon, where Olsen has just kicked off a three-week European tour to promote the album, culminating with three English dates and a Valentine’s Day finale in Glasgow. Last night at Capitólio, a cuboid art deco building in the city’s theatre district, Olsen and her six-piece band made a heroic effort to reproduce the orchestral swoon of All Mirrors in a live setting. The thousand or so audience members may not have noticed a few rough edges in the performance, but Olsen felt the need to account for it. “I’m feeling a little loose tonight,” she confessed at one point in the set. “We haven’t played together for a month.”

The next morning, when we meet at a kiosk on Praça das Flores, last night’s niggles are still playing on her mind. I try to reassure her by saying that slightly imperfect live shows are often more interesting than fully polished ones, but I’m not certain this has the desired effect.

It probably doesn’t help that she’s a little bleary-eyed from post-show birthday celebrations – Olsen turned 33 yesterday and her band marked it by lighting candles in a plateful of Portuguese custard tarts. “We had a nightcap afterwards and then I realised it was three o’clock,” she winces as I order her an emergency cappuccino.

We retreat to a quieter part of the square, away from the traffic, and settle on a weather-beaten bench where Olsen chats freely about the pitfalls of touring and the occasional excesses of her fans, a number of whom have had her face tattooed on their bodies (“That’s really weird to me,” she says). In person, she is warm, funny – and impressively composed, considering the lack of sleep. She tells me she felt nervous last night, but on stage Olsen struck me as imperturbable, even when the entire room interrupted her to sing happy birthday.

Unlike many musicians, she doesn’t seem to mind discussing the (often very intimate) experiences that feed into her songwriting. Later, as we talk about the painful genesis of All Mirrors, I ask if she ever finds herself thinking, in the middle of a full-blown romantic crisis, that she might get some good material from it.

“Oh yeah,” she says matter-of-factly. “And I think that anyone who dares to date an artist or a writer shouldn’t be surprised if they end up in their art in some form.”

Does she ever get people saying: “That was about me, wasn’t it?”

“I’ve had people ask, and then I say: ‘Huh.’” Long pause. “‘It’s interesting that you think that.’” She deploys her finest withering stare. “‘Hmm… I do care about you but… No. It has nothing to do with you.’”

All Mirrors contains plenty of material to get ex-partners in a lather, but right now Olsen’s concerns are focused on the tour ahead. It can be tough going, she says. You don’t know when your next laundry opportunity will be. You end up drinking terrible coffee at motorway service stations. After eight solid weeks on the road in late 2016, she says: “I was not fun to be around. I was drinking a lot. And I just felt so responsible for everyone’s happiness.”

The secret of a successful tour, she has learned, is to keep it short. “Five weeks max. You take the financial hit, so that people aren’t struggling. I really have to consider the people who are playing with me and how [too much time on the road] affects their lives. But I also don’t want to be gone that long. I don’t want to hate playing music.”

Learning hard lessons, and taking charge of her life to minimise future bitterness, is one of the central themes of the latest album. After My Woman came out, as Miley Cyrus et al were queuing up to show their love, Olsen went through a painful breakup with a boyfriend she would rather not name. Around the same time, she started reassessing old friendships that were causing her more grief than happiness.

“I spent a lot of time feeling really bitter and disappointed about the way people in my life saw me or interpreted me,” she says. “For me, All Mirrors is about letting go of toxic relationships, and also forgiving yourself for making mistakes, and not expecting that everyone is measuring you the way you’re measuring yourself. And if some people are, do you really need them around?”

This subject is blown up to near-operatic proportions on All Mirrors’s opening track, Lark, in which Olsen lets rip at a partner who professes his devotion even as he moves to smother her creative ambitions. “You say you love/ Every single part,” she sings over a cascade of wailing strings. “What about my dreams? What about the heart?”

Olsen’s dream of making music, and the single-minded focus she applied to it, arrived early. Adopted aged three by a couple who had fostered her soon after her birth, she grew up in St Louis, a midwestern city she remembers as “kind of depressing”. Her family were poor, relying on her father’s income from the car manufacturer Chrysler, but she had access to guitar and piano lessons – “and I did end up going to rock shows at a really young age,” she says, “so that was fun”.

Her adoptive parents, who already had seven children, were much older – they are now in their 70s and 80s – and Olsen recalls searching for ways to bridge the generation gap. “I started listening to music from the time they grew up in,” she says (the influence of artists such as Patsy Cline and Roy Orbison is often discerned in her work). “I was trying to understand my parents a little bit and think about what they would have been like at my age.”

She also began making music of her own. “I had a little Panasonic handheld tape recorder that I would take everywhere,” she says. “And I was pretty private about it. My mom would say: ‘I heard you working on a song, that sounds nice.’ And I’d be like” – she adopts a stern expression – “‘Please don’t listen. I’m working on it. It’s not done yet.’”

Moving to Chicago aged 20, she enrolled in and dropped out of a massage therapy course, released an EP on cassette, and fell in with folk eccentric Will Oldham, who recruited her to perform in his covers band the Babblers and sing backing vocals for his Bonnie “Prince” Billy alias.

By the time she released her debut album, Olsen’s voice seemed fully formed: she sang about death and betrayal with an intimacy that made her listeners feel like she was confiding in them directly. But with each subsequent record her sound took an unexpected leap. Whereas Burn Your Fire fizzed with a lo-fi rock energy, its successor was more polished, more immediate, veering towards full-on pop on Shut Up Kiss Me, still Olsen’s most popular track (and the one she most resents having to play live).

“For My Woman, I was like, I want to make a classic rock’n’roll record,” she says. “I wanted to have all these crazy harmonies and make it sound like something from the 60s or 70s.” With All Mirrors, by contrast, she had no such plans. “The pieces were coming in one day at a time. It was really scary. I also knew that this material wasn’t as happy or upbeat and it wasn’t going to be immediate with some of my fans who loved the previous record.”

“All Mirrors felt like an end of a chapter in a way,” says Olsen. The album moves from the fury and anguish of Lark to the realisation, on Tonight, that being in a relationship isn’t a necessary precondition for happiness. “I like the life that I lead,” she sings breathily in the opening verse, before delivering the crushing payoff: “Without you.”

A big moment on this path, for Olsen, was buying herself a house in Asheville, the easygoing North Carolina city she moved to in 2013. Having a place of her own to retreat to after weeks on the road underlined for Olsen the pleasures of solitude.

“I joke around about it, but I just don’t really want anyone in my house,” she says. “I don’t want anyone to wake up next to me. In my mind I’d be thinking: ‘I can’t wait till you leave so I can shower and play the music I want.’”

She’s in no rush to get into another romantic tangle – and anyway, she laughs: “The dating scene in Asheville is pretty bleak. It’s like: ‘Did we date already? Or was it my friend that dated you?’

“I don’t know,” she goes on, “maybe there will be some more drama in my life later. But I don’t really feel the need to get crazy drunk any more. I don’t want to go out all the time. I feel boring. My life is calm. And that’s good. But I almost miss the drama like a little bit. You know? Paul, who is playing guitar with me, was like: ‘Don’t worry, something dramatic will happen to you.’”

We get on to talking about politics, where there is an excess of drama at the moment.

“It’s like a TV show,” she says. “It’s like the world is a TV show, with alt-right leaders. And they’re all like trying to be like each other. It’s really strange. But at the same time, I’ve personally seen a lot of younger people become more political. That makes me happy. It makes me look forward to the next election in the US. I have some hope.”

Does she see a Democratic candidate who could deliver on that hope?

“Yes, I think so.” She seems reluctant to specify, so I prompt her. “I mean, I’m for Bernie,” she says. “But of course, we have to keep him alive for a while.”

She sighs.

“It’s hard. And it’s a weird time to be self-promotional about your record and to go on tour and pretend that these things aren’t affecting you on a daily basis.”

But doesn’t music have the ability to help in some way?

“I think it does.” She nods. “I think music has always been used as a way of healing, of encouraging people to take a moment and be reflective.”

One piece of music that affected her recently is by the American soul singer Alice Smith. “My friend showed me her live performance on YouTube, a cover of I Put a Spell on You. It’s insane. It’s been a long time since I saw or heard something like that. It’s really, really special. When I hear performances like that, I’m like: ‘OK, I need to go home and work on my shit.’”

Which brings us to the question of goals. Now that she’s reached the end of a chapter with All Mirrors, and bought herself a house, and a cat, and resolved a bunch of issues and achieved some calm in her life, what’s driving her? What are her musical ambitions going forward?

“I’m just happy it’s lasted this long,” she says.

“I hope to keep growing as an artist, and it’s really cool to see that people want to come and listen to my music. But I don’t really want to be a pop star. I don’t want to merge into that world. It’s fun to collaborate with pop stars sometimes” – she recorded a track, True Blue, with Mark Ronson last summer – “and have a good time, and joke with people about it…”

But in terms of exposure, has she reached a level she’s satisfied with? Is she happy to continue attracting celebrities to her shows without actually becoming one herself?

She doesn’t give it a second’s thought.

“Yeah,” she says flatly. “I don’t know if my label or manager would agree, but yeah. I would be OK with that.”

Angel Olsen plays Bristol, London, Manchester and Glasgow, 10-14 Feb. All Mirrors is out now on Jagjaguwar