The takuhatsugasa (托鉢笠) was a hat worn by travelers to avoid the glare of the sun or protect the face from rain. The hat was made of rice straw woven together. A good portion of the hat shields the face, and if you watch samurai movies, you’ll notice that the hat was used to shield the indentity of the wearer.
The tengai (天蓋) was a hat worn by the komusō (虚無僧) Buddhist monks. The komusō monks are known for wearing straw hats that cover the face. The wearing of the hat is meant to represent one’s desire to deny one’s ego and ambition.