I saw the Art Ensemble of Chicago in concert once, at CASA in St. Louis in 1980, and IIRC they played this tune. I heard it again on the radio once and immediately recognized it from having heard it once before, because of its distinct sound. The main melody is stated solo by Lester Bowie on trumpet. It is a rising pentatonic scale, starting from the dominant. Same pentatonic phrase as at the beginning of Don McLean’s song “Vincent” where he goes “Starry starry night.” The Art Ensemble of Chicago tune I’m thinking of diverged from McLean’s melody after that initial phrase. I don’t remember what key it was in, but if it were in C, it starts in a moderate tempo on the pickup with sixteenth notes G-A-C-D and then downbeat dotted-half-note E. The Art Ensemble of Chicago played it in a “sweet” arrangement instead of their accustomed experimental sonorities.
“Always” by Irving Berlin? Ella did a version of it. Didn’t Jolson have the hit?
It doesn’t really match your melodic description, but “Dreaming of the Master” is based on a mellow swing groove (similar to “Killer Joe”) and the opening muted trumpet solo does kick off with an ascending lick. It is also the only AEC track I can think of that I have ever heard on the radio more than once, and it was in the band’s live repertoire around that time. It’s from Nice Guys, 1978, the AEC’s first album for ECM.