I have both Frank and Back To Black; however, I never dived into some of her demos and remixes. Here are three that stand out.
"In My Bed (CJ Mix)" - A nice use of The Heath Brothers' "Smiling Billy Suite" (famously used by Q-Tip on Nas' "One Love"). It takes the edge off the original.
"Someone to Watch Over Me" - I've been a sucker for Gershwin since I discovered Porgy and Bess; we forget that Amy started out as a jazz vocalist. The first minute and a half are classy, with a hint of the "bad girl" vibe we hear in later albums; it sounds so seductive when she sings the lines: "Only man I ever think of with regret/I'd like to add his initials to my monogram."
"Some Unholy War (Down Tempo)" - The moody backing tracking turns the tune into an elegy of sorts. It fits, especially if you believe that the song is an allegory to drug use and marriage (in effect, Amy's marriage to Blake Fielder-Civil).