For reference purposes I present this traffic police visor cap dated 1939. It is unlined, and a true summer white top visor. The name stamped inside the white lacquered sweatband suggests that it never had a lining not sweat shield. I believe the rivet behind the eagle was the attachment for the front stiffener which has since torn off.

side

Makers stamp inside sweatband
The foam padding has fallen apart behind the white lacquered sweatband.