(Barrie Marshall has immediately offered me Chestnuts Roasting [The Christmas Song] and When You Wish Upon A Star. Robert Duis adds I'm Singing in the Rain. Thanks, gentlemen.)
The only other tune of this kind that I can think of is Bali Ha'i from South Pacific but I doubt whether any traditional jazz bands play that.
I must also mention Jelly Roll Morton's Kansas City Stomp. This famously begins with a series of syncopated rapid octave leaps, though in this case downwards - from the higher note to the octave below. It has proved to be a clever way of producing a dramatic and memorable start to a composition: whenever we hear the first dozen notes, even if the tune has not been announced, our brain responds with 'Ah! They're playing Kansas City Stomp!' (Willow, Weep for Me also uses the downward leap.)