Running Wheel Poop

I've had my chinchilla for a few months now, and I got him a new wheel recently. The new wheel is one of the silent wheels from Kaylee I believe. These wheels have no holes or slots for poop to come out from, so my chinchilla keeps getting it in his fur. Every time he runs I can hear the poop clattering around in there, and it eventually bounces onto his fur. It's not like a ridiculous mess or anything, I just pick the piece or two that gets in it. I'm just wondering if anyone else has had this problem. It's consistent, every time he runs for any length of time he ends up getting some on him.
Incidentally, I do clean the wheel daily, but he poops often enough that, even if there's none in there when he starts running, he usually gets at least one in it.
Has anyone else experienced this and have a good way to deal with it. His cage is otherwise clean, I clean it out daily when I let him out. It's just that **** wheel that's the problem.