That's exactly what I was thinking, the wet dog smell when it rains really annoys me. I have never played baseball at any professional level, but I've played at nearly every other existing level and in a huge variety of locations under equally varying conditions. And I can honestly say that "wet hat" has never been among my top olefactory concerns. Other than the grass, there isn't one single thing in baseball that smells better when it's wet. In fact, grass stains are above hats on the offensive scale. I am willing to bet that when he gets rained on, there are at least half a dozen things on Verlander that smell worse than his hat.
Actually, synthetic fibers are notoriously stinky after extended wear, so I don't really know what the move away from wool is supposed to accomplish in that department. I used to get Backpacker magazine, and one of their primary concerns with evaluating polyester clothing (commonly worn by backpackers for its moisture-wicking abilities) was the clothing's anti-stinkitude. So unless they do a really good job of drying out the hats between uses, they're going to have a bunch of little microbe farms on their hands.
New hats = more $$$$$$$ from the fans.