This style of hat, which was worn from at least the beginning of the 16th century and lasted until it was replaced by short, conical hats at the end of the century, has no name.
Yeah.
I know.
Goddamned Ottomans. They name the hell out of everything else, but they don’t bother to give a hat style a name.
In period, the usual word for a woman’s hat was the generic arakiyye. Modern scholars call the hat a tarpus, a generic Middle Eastern word for a hat that doesn’t appear in any of my period sources.