Are you talking about for field wear, or town wear?I've been looking through "Medieval Life Illustrations". Since none of these woodcuts are credited, it's difficult to know what country they are from, but it's safe to assume the Continent. A note in the front of the book says that many of the illos come from "Der Bildersmuck der Frühdrucke" but that's all.
For "field" wear, I'm seeing a lot of these pleated deals on the women- I would LOVE to know how they're accomplished! They look as if they might be a variation of the Fram air filter hat on ven der Wyden's Magdalene. Maybe I need to sit down some weekend and work on it, as my attempts have been unsuccessful to date. If anyone has a good headwrap pattern, I'd love to see it!
You could always do a scarf kind of deal as the layer closest to the head and then cover it with a veil or veil and gorget. Some of these look as if the veil is set on the head with most of the fabric hanging off one side forming a tail. The tail is then brought under the chin and either tucked or pinned so as to cover the throat. That one seems very popular. A similar style is sometimes seen with a much longer tail brought across the top of the bosom and thrown over the opposing shoulder, much like the liripipe on a chaperone.
I'd say avoid any sort of structured hat for field wear, and avoid a simple scarf/coif for more formal wear. Even a veil and gorget looks to be Ok worn with an over dress for town wear.
Gwen