That's a really deep, big honk and rich tone in the sax, and the sax leads the head (first time through) the song really strongly, and it's a smaller combo ... I'd say dig through later Illinois Jacquet recordings and see what you can find. His solo on the original hit with Lionel Hampton was his #1 claim to fame, and I think he milked it a lot (there's a later recording of his I play quite a bit that's basically a rewrite of Flying Home).
Otherwise, when I hear honking sax as the featured artist & intense jump-blues-ey swing, I think Louis Jordan.
Beyond that, there are some celebrities of jump blues and later Rock&Roll who had very intense sax sounds who might've banged out a swing tune ... I'd be surprised if this is someone we've never heard of on that sax, though -- despite the low-quality recording.
And my first instinct says, "still Jacquet."
Update: A search of "illinois jacquet" "flying home" piano drums
turns up this:
https://forum.saxontheweb.net/showthrea ... ost2578456
-- there's a 1944 (late enough to be intensely jump-blues-ey, early enough & during-the-war enough to be badly recorded) Jazz At The Philharmonic concert where Illinois Jacquet played in a sextet with Nat King Cole on piano -- I like that combo as a possibility (the piano sounds really prominent & skilled, too -- like someone famous -- but also someone with some notion of an influence from Basie, given the moments without a lot of notes -- maybe Oscar Peterson ... he seems to have played with Jacquet a bit, too, but King Cole sounds like a good lead).
See if you can get ahold of what seems to indicate "disc 9, track 7" of a Jazz At The Philharmonic 10-disc box set. This lineup sounds promising, except that the bigger names than the pianist aren't really featured enough...
https://www.discogs.com/Jazz-At-The-Phi ... ase/476803
Alto Saxophone – Charlie Parker
Bass – Ray Brown
Drums – Buddy Rich
Piano – Hank Jones
Tenor Saxophone – Flip Phillips, Lester Young
Trombone – Tommy Turk
Trumpet – Roy Eldridge
Nope ... I take it back ... that seems to be https://www.youtube.com/watch?v=c3MGOigwz_w