Amy Winehouse Lioness: Hidden Treasures includes a track recorded while the troubled star was under the influence of heroin.
Another song in the surprise collection — out in December — tells of cheating by ex-husband Blake Fielder-Civil during the couple's turbulent relationship.
Amy's emotional dad Mitch wept as he listened to the album — and admitted: "I never realised what a true genius she was."
The ex-cabbie could not hide his pride as he listened to the 12 songs, some of which even he had never previously heard.
I joined the family in a West London studio last week for the first listen to the surprise collection.
Wiping away tears of joy Mitch, 60, said: "I spent so much time chasing after Amy — telling her off — that I never realised what a true genius she was.
"It wasn't until I sat down with the rest of the family and listened to this album that I fully appreciated the breadth of her talent. From jazz standards to hip-hop songs, it took my breath away."
Proud ... Mitch Winehouse with daughter Amy
Amy's 56-year-old mum Janis listened separately to the collection of unheard material because the emotion of the occasion was too personal to share with others.
Mitch added: "One of the new songs on the album, Halftime, I'd never heard before. It is just incredibly beautiful.
"Had the family felt this album wasn't up to the standard of Amy's others, Frank and Back To Black, we'd never have agreed to release it. We believe it will stand as a fitting tribute to her musical legacy."
Producers Salaam Remi and Mark Ronson, Amy's close pal, have been locked in studios across the world for the past few weeks pulling the album together.
They shared the heartbreaking task of listening back to thousands of hours of vocals by the singer, whose troubled life was cut short in July at the age of just 27.
Salaam said: "When I listened back you would hear some of the conversations in between — that was emotional. It has been hard, but it has also been an amazing thing. Amy was a gifted girl.
"I believe she has left something beyond her years. She has put a body of work together that will inspire an unborn generation.
"I'm blessed to be part of that process, to have known that person and to continue her legacy with this album." Amy, who scored a string of hits and won five Grammy awards despite battling drink and drugs, died at her home in Camden, North London.
An inquest last week heard she was five times the drink-drive limit when a bodyguard found her slumped dead on her bed. No drugs were found in her system when she died and she had been clean of them since 2009.
Legacy ... cover of new album Amy Winehouse Lioness: Hidden Treasures
The posthumous album will be released on December 5 and £1 from every sale will go to the Amy Winehouse Foundation, which is helping children's charities.
The most emotional track for the singer's family and fans is her cover of A Song For You, a Leon Russell tune made famous by Amy's favourite artist Donny Hathaway. The song, on which Amy plays guitar, was recorded in a single take at her London home in 2009 as she battled to overcome her addiction to Class A drugs.
Her voice can be heard quaking as she tearfully delivers the lyrics.
Another cover, of Carole King's Shirelles classic Will You Still Love Me Tomorrow, was produced in 2004 — months before Amy embarked on her drink and drug hell.
And a further track, Between The Cheats, was recorded as the star emerged from her turbulent marriage to Blake Fielder-Civil.
'Artistic genius' ... Amy Winehouse with Tony Bennett after recording Body & Soul
Others include Wake Up Alone, from 2006, and classic The Girl From Ipanema — recorded in 2002.
The last song she ever recorded, Body & Soul with Tony Bennett, also appears. The session, at Abbey Road Studios in London in March this year, marked one of the happiest days of her short life.
Tony, 85, said: "It was a thrill to record with Amy Winehouse. The recording is a testament to her artistic genius and her brilliance."
Proud Mitch poignantly recalled Amy's amazing lack of ego.
He said: "I remember taking her shopping at a particularly difficult time in her life a few years back.
"We looked through all these racks of clothes, she picked something out and I bought it for her.
"As we were waiting, one of her songs came on, I think it was Stronger Than Me. She just looked at me and said, 'Is that one of my songs, Dad?'
"She genuinely didn't listen to her own music after it was recorded. She wouldn't wallow in her success. That's what was so lovely — she was just Amy. No ego about her."
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