Jazz St. Louis is the city's main dedicated jazz room, set in the Grand Center arts district a short walk from the Fox and the Sheldon. The club books serious local and touring musicians in a listening-room setting, so it works best when you want an actual concert rather than background music with dinner.
St. Louis has a deep blues and jazz history, and this is one of the easiest places to hear that tradition treated with care. Check the schedule before you go; the strongest nights sell out.