The Jayhawks with Joe Henry.

I believe that I recall reading that the band first played with Dylan at the Newport Jazz Festival. Dylan was just beginning to go electric and lots of folkies were booing him live. I'm sure playing with Dylan opened some doors for them but as someone said they were playing as the Hawks beforehand. Levon Helms actually left them when they decided to back Dylan up as he wasn't interested in being in anyone's back up band, and rejoined when they left Dylan. The actual name "the Band" didn't come until after they signed with Capitol. They originally signed as the Crackers which Capitol made them drop once they figured out what it meant.

Others:

Teenage Fanclub has backed up Jad Fair in addition to Frank Black
The Bar Kays backed up Isaac Hayes
Black Merda Backed up Fugi
Trio Mocoto Backed up Jorge Ben

Feel like there should be lots of soul/funk examples but I'm drawing a blank

And the new album I just picked up today is a return to form............