What's wrong about the following statement? "During the Middle Ages, all farmer's maids used to wear big, pointy hats that looked like cones." It's poppycock, of course.
Which is exactly why you'll find a variety of different authentic head pieces from 'back in the days' - among them warm gugels, decent bonnets, barets and hats made of linen, felt or precious velvet.
The pointy hat, however, did exist. It's called a 'henin', or 'burgonet' (but only in the 14th Century). We have these, too. And to make your outfit complete, you will even find matching garments and dresses. Have fun browsing!