I have a set of Pirelli Wets on a set of rims, ready for damp conditions. These are the 'inter' tyre from Pirelli (as distinct from the Rain tyre, which is confusingly their wet tyre). I haven't actually had the opportunity to try them yet, but they look and feel like the goods!

The other great option for an intermediate tyre (most race guys do this) is to use old wets that are a bit torn up. They are perfect for the damp, and it doesn't matter if you destroy them!

Road tyres are next to useless on track in the wet/damp. You might as well be on slicks.