You are probably thinking of Limehouse Blues. It is a song recorded by many different artists, Django Reinhardt being one of them. If you listen closely you can actually hear the character cricket say "okay, Limehouse Blues" just before they start playing it.
I looked at the soundtrack listing for To Have and Have Not on IMDb and Carmichael is credited playing Limehouse Blues, but a studio orchestra undoubtedly backed him.
Reinhardt influenced so many jazz guitarists. I thought a movie of this stature would have attempted to entice Django to play the music.
Django Reinhardt was living in occupied Paris during the war, when this movie was being made. He had been unable to escape France, and as a gypsy and a jazz artist he was in constant danger of being sent to a death camp by the Nazis. Only the protection of a few sympathetic Nazi officers who happened to be fans of his kept him alive at all. With jazz being officially outlawed and gypsies being marked for extermination, trying to get passage to the United States to make a cameo playing jazz in a movie where Nazis were the bad guys would have been neither easy nor healthy.
Did Reinhardt with his group appear in any feature film, anywhere in the world? His band's sound was so unique, a performance could easily be inserted into a film.
I'm not aware that he ever actually appeared in any feature film. There have been documentaries with some clips of him playing, but that's all I've ever run across.
Funny this thread is here...when I heard the quitar solo in question I immediately wondered if it was Django. I backed it up to check...nope, no Django in Cricket's band.