How about a holster shirt? There are a few manufacturers...
511 Tactical
A Better Holster
It looks like something you could wear under a running shirt, sweatshirt, wind breaker, or the like. Only problem I see is that it wouldn't provide a moisture barrier for your firearm. Though I would think a good wipe down with a lightly oiled gun rag would be all you need to protect your firearm.
Fly