The subgenre is called "bebop." Get some Charlie Parker albums. He recorded some great songs with Dizzy Gillespie. Jet is talking to Spike about talking to Charlie Parker in a dream in the casino episode, iirc. The style of music is fast tempo, quick key changes, novel chord progressions, and virtuoso performers making new music out of standards. It's analogous to the storytelling in the manga, and to the characters themselves. Each is supremely competent, acting on their own, but complementing and supporting the others to make something extraordinary. The whole soundtrack is a wide range of genres, and it was all written and performed by Yoko Kanno and the Seatbelts, which is especially impressive because of the sheer variety of styles.

Tank! is more driving trumpets and melodic than classic bebop, so you might also check out some Wynton Marsalis. He played what is called "neo-bop" which was a popular revival of bebop in the 1980s.

Jazz aficionados would probably classify Tank! as "hard-bop" of which there are many great albums and musicians. John Coltrane's A Love Supreme was one of my favorite albums growing up, but that was the tail end of his hard-bop phase. I would probably suggest Art Blakely and the Jazz Messengers album "Hard Bop" as the quintessential hard-bop album.