First step (granted perhaps a little late now) is to bring that hat with you when you buy your helmet and make sure it fits comfortably under your helmet.

I like simple wool hats that can have a fold at the ears or the edges pulled down further. I find wool much better than acrylic. I wear (and size my helmets for) a Pace of Halo cap under whatever winter hat I use. Hats stay cleaner and my hair feels cleaner. (I have a bunch and always leave with a clean one.)

To get a step more warmth, I use one of those rain covers that fit over the helmet, blocking the vents. Big step up in warmth. I try to arrange it so it is held by velcro and is easily removable when I warm but don't always succeed.