The 30-year-old singer, who appeared alongside Johnny Depp and Kate Moss in Paul's recent video for his track 'Queenie Eye', was given some tips from The Beatles legend and has kept in touch with him.
Jack told BANG Showbiz: ''I was very lucky to be able to hang out with Paul McCartney for two hours and he's just an absolute legend.
''I was surprised by his enthusiasm to talk about the stuff he's probably been asked [about] a thousand times. It's was as if I was the first person ever to ask him, 'What was it like to be a Beatle?'
''He asked me how things are going and he said, 'Don't worry about things too much, just make sure you come up with good songs'.
''He played me Lady Madonna on the piano and none of my friends believed me. It was a great experience and he has kept in touch since. He sent me a card on my birthday and a Christmas card. I wasn't expecting that.
Jack, who will release her new EP 'Sweet Hurt' on March 24 on Lanza Records, also previously recorded a duet with pal Sienna Miller called 'Love and Hate', which featured on The Vampire Diaries.
He said: ''She's an incredible singer. I saw her at the Groucho Club [in London] one night stand up on a table and sing Nancy Sinatra acapella. I remember being completely shocked like, 'I didn't know you could do that'.
''She was going through something, I was going through something [at the time] and I wanted to write a song about the fine line between hate and love.''
Jack, who has released three albums, is also about to step out on a headline tour of the UK, after supporting the likes of Jake Bugg and Bruce Springsteen.
He added: ''I think the crowds in the UK are actually the most knowledgeable about music, you have to pick up your game. You have to give them something they haven't heard before or haven't seen.''