True more apt to costume, but to give a brief answer here, it depends!!
Firstly it depends on which bit of medieval you're talking about - social attiudes did change through the thousand or so years that can be called medieval (obviously).
But it also depends on where you were (although that's hard to pin down unless you have a text or pictures directly relating to a given area.
And finally it would depend on the age of the woman. Young women, no - not until they *were* married. Older women who weren't married (and hadn't turned to the church), may have covered their heads just to protect themselves from attacks brought on by local moral outrage.
"What a lovely hat! But may I make one teensy suggestion? If it blows off, don't chase it."
RIP Edward the avatar cat.