Place the chatbox on top of the records screen post-game.
Topic title syas it all. Post game chat is very hard to read because it's hidden behind the (sensibly, normally) 'obtrustive' stats screen. But when you're engaging in a bit of post game chat it becomes very hard to read what people are saying. If the box floated on top of stats when you popped it up to type, it'd be a lot easier to see what you're saying, and what others are saying.