Amy Helm interrupts her conversational flow with foreboding.

She's approaching a toll road and has managed to leave the house without any money. Being the daughter of the late Levon Helm, a local hero around Woodstock for more than 40 years, will exert influence and open doors in many circumstances in the area but the singer who is following in her father's footsteps is on her own with the toll operator.

An effusive apology, a winning explanation and an IOU that will allow her to settle up at the next opportunity and she's back on course, picking up the story of how she and her musician friends are keeping up the Midnight Rambles, a series of informal gigs by musical luminaries, which her father established at his Woodstock home.

"We're honouring him and I think doing what he'd want us to do in creating opportunities for music to be played and listened to and for people to have a good time," she says, adding that bringing her own music to Celtic Connections' equivalent of the Midnight Rambles, the Roaming Roots Revue, is a thrill almost beyond words. "A lot of my friends have played in Scotland and they've always come back raving about what a special experience it is. So I'm looking forward to experiencing it for myself."

She hadn't intended going into the family business – as well as her dad being the master drummer and one of three distinctive voices that brought The Band's music to dancing, swinging life, her mother, Libby Titus, is a songwriter who is now married to Donald Fagen of Steely Dan. Helm grew up in Woodstock and Los Angeles while her father made his inimitable contributions to songs such as The Weight and The Night They Drove Old Dixie Down and toured and recorded with Bob Dylan. Then when her parents split, she spent weekends with Levon in Woodstock but went to school in New York City, going on to gain a degree in psychology from the University of Madison, Wisconsin.

"I always felt comfortable singing in choirs and doing school shows but maybe through seeing my dad struggling at times after The Last Waltz [the star-studded concert-film that was intended to lay The Band to rest but reckoned without its drummer's determination to carry on] I ran away from music for a while," she says. "But when I decided to give it a try. Boy, my dad was very insistent that I get as much gigging experience as possible and I learn as much about the music that's his legacy really as I could possibly take in. He was amazing, a real force of nature, and I'm really grateful he took me under his wing."

Her early experience as a professional musician was gained from the late 1990s through to the early noughties with her dad's Barn Burners, wherein, she recalls, there was no getting away from the fact she was the great Levon Helm's daughter.

"It was scary having to live up to people's expectations," she says. "But I just had to be myself. I practised and prepared for gigs like mad. I didn't want to let my dad down but he was so supportive and encouraging and that helped me begin to feel comfortable and really enjoy what I was doing."

The onset of throat cancer robbed her dad of his trademark, soulful, stoic yelp but he carried on playing drums. "Even when his voice struggled he could still outplay everyone," she says. "And he had this great understanding of where the drums should be in any song. It was really inspiring."

Working on the Dirt Farmer and Electric Dirt albums that gave her dad, once his voice had returned, a late life revival and Grammy success was, she says, like going to university. As they worked up lists of possible tracks, her father dredged up memories of songs his parents had sung around their Arkansas farm when he was growing up, revealing a treasure chest of folk-country-gospel lore that added to the blues and early rock'n'roll he'd often concentrated on in later years with his own bands.

"It's an incredible legacy to have on tap," says Helm, "and I've taken so much from him into my band Ollabelle, which is still playing, and my solo album, which I hope to release this summer. Although he died last April, I can still feel him with me, giving advice on how to sing a certain line or just keeping at me to work, work, work. It's funny because when I was young, I thought my dad played in Rick Danko's band because the only Band song I knew was Stage Fright and Rick sang that. But of course it turned out my dad was a great band leader and he knew so many songs; his knowledge of American music was encyclopaedic and I'm so glad he was able to share that with me."

Amy Helm appears at 02 ABC, Glasgow, on Friday and at Glasgow Royal Concert Hall on Sunday as part of Celtic Connections.