Is it a bonnet like the pioneer ladies wore? Snug around the back of the head but with a big brim?

It is not like the pioneer bonnets. It is made of net or chiffon. goes over the head, is attached to something that ties in front or under the chin.
ggks

It is not like the pioneer bonnets. It is made of net or chiffon. goes over the head, is attached to something that ties in front or under the chin.
ggks

Try the link. They call them a Wind Cap.

They are like those cheap plastic folding rain hats, but of chiffon or a soft netting. You could take a rain hat apart to use for a pattern.

Here is a link that might be useful: Wind scarf or bonnet

You might try a Sally's Beauty Supply, or a local drug store.
Kathy G in MI

Based on the plastic (for rain or wind) ones my mom used to carry/use, I think you could do the following (prototype first from some paper or plastic):
-measure your head from neck to forehead--call this the "height."
-measure around your head from chin to chin (not trying to imply you have 2 :-) --call this width
-cut out this size and fold it back and forth (fan-style), so that its height "collapses" to about 1 or 1 1/4 inch, and its width stays the same. (i.e., draw your fold-lines from "chin to chin"); for fabric, you will need some method of retaining those folds.
-with it folded, sew (or staple, for your prototype) on ties to each end, capturing the folds so that each end will be permanently folded. make ties 9 inches or so long (each; as a rule, you need at least 18" to tie a bow)
-try it on, opening up the folds. if your fabric is holding the creases, when you take it off, you can pull on the ties and it will collapse back into the fan-fold so that you can store it in a very small pouch.