Thanks for all of your inputs!

I have to admit I never would have considered the diver's helmet as a sun block option, but since I commute along a river part way, it might come in handy if I were to fall in. Just have to make sure that the air hose is long enough and does not get kinked. I am not sure about the umbrella hats though.

The Headliner Helmet Cover might work out. I also saw an instructables article which details how to adapt a brimmed hat to a bike helmet by cutting off the top off the hat to provide ventilation and cutting slats through the brim for the helmet straps. See htttp://www.instructables.com/id/A_DIY_Sun_Protective_Brim_for_a_Bicycle_Helmet.