The ideal winter walking hat has a bill to shade your eyes and keep the rain and snow out of them while insulating your head and ears to keep them warm.
Hat hair potential: High. Even in the cold, your head can steam and sweat, and most winter walking hats will crush your hairdo.
My favorite winter walking hat is the Sub4USA DRYLYTE Extreme Cold Hat (pictured). It has a bill and it has a built-in ear band to keep your ears warm. Unfortunately, that company has gone into retirement and it is no longer available.
Your basic beanie hat works fine for winter walking if you don't mind sun and wind in your face. It does the job of keeping your head and ears warm.(Compare Prices)
As it gets below the freezing point, you may want to add a neck gaiter or balaclava. (Compare Prices