|British Library Stowe 17|
It took more than dusting. My sources in 2006 were books, including the "history of costume" books by Norris, Planché, Köhler, and Houston, and several less useful but more modern ones. This time I was able to visit websites of the Morgan Library, the British Library, national libraries of the Netherlands, France and Austria... I lost count of how many. With their online publications I was able to pull together about 1000 pictures of 13th and early 14th headgear, focusing on the one worn by the model in the picture. The pictures included several media, including cathedral statuary, which aided interpretation of the painted media.
This headwear, known in the literature as a coif, touret, turret, filet, pill-box hat, pie-crust hat, and coffee filter hat, had several variations during the period of its popularity (between 1183 and 1416). What dating was available provided a nice bell curve, peaking around 1250. The sides of the hat might be parallel or flared to varying degrees. It might be tall enough to hide the top of the head, or the head might poke through. The hair might be braided, and fastened behind the neck, or, more frequently contained in a solid or net caul. It was almost always white. The hat was almost always worn with a barbette, or chinstrap.
Though the 500 pictures of the "real hat" (as opposed to similar-looking crowns, or other types of head covering) are hardly a scientific sample of all of the evidence, I feel very confident that the people in my class have a clear idea of what "the 13th-century hat" looks like. (And I now call myself an SCA CSI.)