I live about 1 1/2hrs from there and grew up on the Illinois side of the greater St. Louis Area.
While I can't make it down there in the next week, I can provide any intel you may need.
I would recommend looking at the "Hyatt at the Ballpark" or even the Drury Plaza Hotel at the Arch. Both are right in the middle of everything and nice hotels. You're going to want to hit Ballpark Village, and it is literally across the street from both.
I haven't been there for a smoke honestly, and can't think of any places off hand, but I'll look around.