ToniG wrote:The works of Nelson Algren, specifically The Man with the Golden Arm, whose protagonist is a jazz musician. Studs Terkel's books, and most especially his new one, And They All Sang, a collection of his interviews with musicians. (No one would know more about the specifics you're looking for than Studs, whose radio show began in the 1940s.) And visit the archives of the Chicago Historical Society (where the archivists are very helpful) if you really want actual primary information. I love website convenience too, but real history doesn't live there.
Frankie Machine is hardly a jazz musican...
Here's some more stuff I dug up:
Patricia Bronte's Vittles and Vice is from 52 but otherwise on point.
Ian Flemming (of James Bond fame) wrote a series of "Trilling Cities" in 59 and 69 for the Sunday Times. They were collected then in print. One of the cities is Chicago. Again, it is a bit post dated for you, but a lot of the stuff in the book would apply to the 40's, at least to the late 40s.
And speaking of late, but on point, Ovid Demaris's Captive City is a much better version of the material covered in Chicago Confidential cited above. It is from 69 though. In the same vein Kup's Chicago was written in 62 but covers the 40's fer sure.
There are 3 or 4 biographies of Al Capone out there. A less known mobster of that era is Murray the Camel Humphreys. There is an interesting book about him by John Morgan.
Have fun!
Think Yiddish, Dress British - Advice of Evil Ronnie to me.