I was just at Juicy Wine Bar and, although I didn't have any cocktails, I am pretty sure that they had some on the menu. They also have a nice (but small) outdoor deck area. If you want to talk, you might want to sit there since the main room can be a bit loud.
The one time I went to Paramount Room it wasn't very loud. In fact, it was pretty empty.
Another option is the Iguana Cafe. I don't think they have cocktails, but I could be wrong.
Finally, Matchbox is close by. They have great cocktails. The outside seats are very conversation-friendly.
Iguana Cafe (and wine bar)
517 N Halsted St
Chicago, IL 60622
iguana-cafe.net
Matchbox
770 N Milwaukee Ave
Chicago, IL 60622
(312) 666-9292