As I place two full-to-the-brim pints carefully on the table, I realise I'm whistling a tune that has been trapped in my head for the duration of the day so far.

It's only when I glance up at my companion in the back room of this pub in Edinburgh's New Town, and see Dan Willson, aka Withered Hand, sitting opposite me that I recognise it as the catchy verse of Horseshoe, current single and opening track on Withered Hand's forthcoming album, New Gods. Now, that is a wee bit embarrassing…

Or maybe not. It's no secret that I'm a fan of Willson's work. He tells me I'm the only person who has ever given one of his releases a five-star review ("this is the album Neil Young wishes he could still make" is what I said of his debut, Good News, in the Sunday Herald in 2009). And I'll happily go on record now to say that New Gods might be even better: it's certainly a more confident set of songs, with a bigger band sound and tunes that bask in the glow of glorious pop music.

"When Good News came out, it surprised me that people got into it because I hadn't really known there was such a universal aspect to writing about things which are quite specific," Willson admits. "Then I realised that's what I like in songs. That's what we all like in art. Someone does a thing that's specific to them and it resonates across a whole lot of people, and they don't feel like they're living in isolation."

It's not surprising that Willson uses art as a wider example when talking about music. Growing up in Bishop's Stortford in Hertfordshire (where he was raised as a Jehovah's Witness, thus providing plenty of lyrical material for his debut album), he initially wanted to be a visual artist and did try to give it a go after moving to Edinburgh to be with his then girlfriend, now wife, in 1996. His art endeavours have, however, fallen by the wayside.

"It's sadly neglected because of the music and because of my lack of success," he admits. "My lack of success in expressing myself visually, I guess. One time, when I was a cleaner in a gallery, I was thinking, 'This is wrong. This is the worst thing I could do. Cleaning the toilets in an art gallery when I want to be an artist is probably the way to forget your dreams'.

"I kind of sailed through art school, and I've always been good at drawing and stuff, but was never really able to channel myself into it. That's why songwriting became so liberating for me because, in a way I hadn't expected, it did the thing I'd thought art would do for me: liberate me and give me a voice."

It's a voice that has clever, self-deprecating things to say about the spiritual tussles in his inner life, but it's not a voice that likes to shout from the rooftops or hog the microphone. Willson would be the first to admit that, when Good News was released, he could barely consider himself a singer, having only begun writing and performing on an acoustic guitar when he received one for his 30th birthday. He's not exactly a reluctant musician, but he'd certainly place his job as a dad to two children high above any demands the music industry might make of him.

"If I'm away from home on tour for more than seven days, I need to be pointing in the direction of home, or I'm not a happy man," he says. "It'll be all right if we're playing another seven days, as long as we're heading back the way … The key is that you just bite off what you can chew."

New Gods is such a good album that Willson might find his Withered Hand workload increasing rapidly. The album will be released later this month on the Fortuna POP! label in the UK and Slumberland in the US; he's also heading to the South By South West festival in Texas in a few days' time as part of the official Scottish Showcase supported by Creative Scotland. Hopefully the trip will prove a less stressful experience than the one he endured in 2011, when his visa was denied until he could prove (with the Scottish Government eventually backing his case) he did indeed have "extraordinary ability" and had achieved "significant recognition". As diplomatic incidents go, it wasn't the Cuban Missile Crisis, but still…

I reckon there is a natural constituency for Withered Hand's music in the US. Beneath the more exuberant indie-pop sheen of some of the tracks on the new album, the influence of Neil Young is never far away. In fact, New Gods' cover artwork perfectly captures this - and Willson's ambivalence about it. There he is in a checked-shirt, in a guitar-totin' pose steeped in the Springsteen/Young tradition; but, at the same time, he's ducking out of that lineage, face-palming himself and obscuring his eyes. A cynic might look at the track listing and spot a few songs that seem, from their titles and lyrical locations, to be pandering to an American market. Not so.

"I wrote those songs as processed experiences of the horrible time I had there," Willson explains. "There's a kind of travelogue across three songs. Love Over Desire is me leaving America, leaving the people I love to go somewhere else and do something which is of dubious merit. King Of Hollywood is finally, after many experiences, hooking up again with King Creosote in LA and being, like, 'A friend!' And California … well, I was taking the wrong medicine, basically, and it was making me panicky and unwell. Those three songs had to go in that order and they had to go on that album."

His reference to King Creosote leads us to talk about how the man himself, Kenny Anderson, has been something of a mentor to Willson, and how the Fence Collective provided a sense of support that Willson has felt generally from the indie music scene in Scotland.

"It's important to have contact with people who have been through the mangle and know a bit about what you do. Otherwise you can make so many mistakes. I haven't asked Kenny too much, to be honest, but there have been a couple of times when I just needed to know nuts and bolts stuff - am I being diddled here? - and I'm really lucky that between him and a few other people, I've got some wise old ears to ask."

There was another set of wise old ears listening closely to New Gods as it was being recorded: producer Tony Doogan, who has worked with Mogwai, Teenage Fanclub, The Delgados and David Byrne.

"I chose him for a reason," Willson admits. "Those early Belle And Sebastian records he worked on are really big in my life. Tony's involvement was crucial to get the record sounding like it does."

The only time Willson gets insistent as we talk is when it comes to giving credit to others. He's keen to flag up the contribution made by the friends and musicians who play on the new album: cellist Pete Harvey and drummer Alun Thomas (The Leg), guitarist Malcolm Benzie (eagleowl), Eugene Kelly (The Vaselines), Scott Hutchison (Frightened Rabbit), Pam Berry (Black Tambourine), Peter Liddle (Second Hand Marching Band) and of course King Creosote. A few days after we meet, he contacts me on Twitter to add a few more: Chris Geddes and Stevie Jackson (Belle And Sebastian), Benedikt H Hermannsson (Benni Hemm Hemm), his own kids who do backing vocals. All in all, it's a fine cross-section of Scottish indie and proof that this man from Bishop's Stortford is now one of the scene's adopted sons.

"I really do feel part of it," he says. "I don't want to talk about independence, right, but in the future, if I had to apply for citizenship in a fantasy world, I would. The best thing I ever did was come and live here. I feel like an ambassador for Edinburgh. Whenever I go anywhere, I'm just raving on about Edinburgh. It saddens me that I wasn't born here, but what can I do?"

You can give us your songs. That is payment enough.

New Gods is released on March 10. Withered Hand plays a solo set supporting Second Hand Marching Band at Mono, Glasgow, tonight, and full band shows at Liquid Room, Edinburgh on April 17 and CCA, Glasgow on April 18.