If the hat were a later addition, it would make our job a bit easier, but I'm almost certain it isn't. The linework looks quite integrated with the rest of the drawing.
About clothing terminology: if we know the proper names it might be easier to find help in the literature. For example, now we know the hat is one of two options:
- a chaperone without the usual frills, so it would just be the "bourrelet" part
- a completely different hat type that looks like a bourrelet but did not evolve from a hood like the chaperone did