Two possible solutions: a new hat that you can spray with waterproofing, or a kiddle tube.
Most fishing hats are quick-drying and can easily get wet without being ruined. You can even get the kind with a neck scarf that will keep your neck from burning. If you heavily spray a fishing hat with waterproofing it will even keep your hair dry underneath it.
The simpler solution, unless the place is truly packed, is to snag yourself a kiddie tube in addition to the one you're sitting in. These have a solid bottom, and if you bring a cord-loop with you to put around one handle, you won't even have to hold it most of the time. As you near the bridges, just pick up the kiddie tube and hold it over your head upside-down until you've passed beneath.