The secret link in Jacquet is Lester Young. There's a distinction between "White Lester" and "Black Lester" in terms of which general way his influences landed.
Personally, my preference for Jacquet is from the middle 50s on. It seems like he (mostly) set aside the JATP crowd-pleasing devices and began just playing, like on the track here.
No matter, the guy was a superb player, period. There's a rich body of work to explore, so have fun with it!
Oh, and he played very credible bassoon as well!