Parts:
<-cross-section
The plastic mesh stuff is very non-period, but it made the whole thing very easy to construct. A lucky find in Spotlight. Originally approximately A4 in size, cut in half and sewn together, then wrapped in cloth and a cloth top attached.
Couldn't find any images of excatly what I'm trying to reconstruct online, but it is similar to this pattern by Marc Carlson. A variety of pileus rotundus.