The previously-unheard tracks feature on Lioness: Hidden Treasures, an album compiled by Winehouse's producers Salaam Remi and Mark Ronson and approved by her family.
'Our Day Will Come' was recorded in May 2002. Winehouse's version of the '60s pop classic gives the song a reggae twist.
Listen to 'Our Day Will Come' below:
The track aired on Radio 1 and breakfast show host Chris Moyles said: "I think if you're an Amy Winehouse fan, it's going to be a bit of a tear-jerker... When you hear her sing now, I think you hear it differently."
His co-host Dave Vitty also paid tribute to Winehouse's "haunting" voice, comparing 'Our Day Will Come' to a song from a film soundtrack.
'Like Smoke', a collaboration with Nas originally recorded in May 2008, aired on BBC Radio 1Xtra. Before he played the track, host Twin B hailed Winehouse as "one of the most talented, original and best popular music artists to emerge in decades".
Lioness: Hidden Treasures features a selection of demos and rarities spanning Winehouse's musical career.
Remi - who worked with Winehouse on both Frank and Back to Black - has promised that the collection is "faithful" to the late musician's legacy.
Lioness: Hidden Treasures will be released on December 5, with £1 from each sale going to the Amy Winehouse Foundation.