Women were not "required" to wear hats during the regency period as they were during other periods in history. Plenty of images show women running around with headgear. Hats were more a pretty adornation, like some men in modern times feel they must wear a baseball cap :) Also, they of course kept the head warm on cold days!
Regency Period and Jane Austen
Hats of this period were a "scoop" shape, similar to the scoop one would use to gather up coal. Their narrow focus could be extreme enough to hide the wearer's face - which of course served its purpose of keeping a sharp wind off of one's face.
Elizabeth Bennet from Pride & Prejudice BBC version -
Four of the girls from Pride & Prejudice A&E version -
Charlotte Lucas from Pride & Prejudice A&E version -
Lady Lucas from Pride & Prejudice A&E version -
Women were often doing and redoing hats using the same base. So they would have the base shape in whatever material they chose, then they would add on ribbons, bits of lace, flowers, and so on based on the outfit they wished to wear.
Regency Period Information