For rain, you have two ways you can go. First, you can wear a ball cap under your jacket's hood. This can be nice when a hat with a brim might interfere with something else, such as a pack or bowstring. The caps bill helps keep the hood out of your way and keeps the rain off your face. It also provides the most protection. The downside is that the hood can interfere with your hearing and possibly your vision.
Second, you can wear a brimmed hat. This give good coverage without a hood and doesn't interfere with your hearing. If its really raining, I use my Seattle Sombrero. If its not too heavy, my vented nylon Tilley is great.
-- FLIX