When Karl Denson played at the Jazz Fest a year ago, he was on the main stage with the Rolling Stones. The saxophonist and flautist has been one of the rock band’s touring musicians since 2014 when he replaced the Stones longtime horn man, the great Bobby Keys.

But Denson’s no stranger on the other stages at the Jazz Fest. He has led his own band, the Tiny Universe, four times begin...