If you wear a traditional bonnet, not the beret'ish amalgamation you can pull it forward to shade your eyes, its pretty much a flat cap.

Then there are the billed headwear of the 19th century, feather bonnets and even just diced bonnets with bills on them.

We need a new word, since "Traditional" has the generally accepted definition of doing something the way your grandparents did it, and Im usually doing something the way my 5th to 7th Great Grandparents were doing it LOL