The reason "disco" came up in your search is there is a very popular disco cover of this particular song. You are correct, however, that the original style is not at all disco.
I've seen the genre of this song described as "ragtime," an early synthesis of African-American syncopation and sensibilities with classical musical structures --and you will find that ragtime does in fact have the "upbeat bounce" you are looking for. However, ragtime is primarily associated with solo piano. The adaption of ragtime to big bands created a genre later called "Dixieland Jazz" or "hot jazz," which is more directly what you are looking for.
As Bebs correctly noted, this is often called "New Orleans Jazz," since it was really the first style of what we now call jazz, and New Orleans was its birthplace. However, since New Orleans is known for many many different musical styles, "Dixieland" or "hot jazz" are probably your best bets for a useful search term. ("Gipsy jazz" does have a similar sensibility, but tends to focus on guitar and violin, not wind instruments).