Everybody wore hats, and I mean everybody. Men, women, kids. Hats were normally worn in the streets, but not inside. They shielded people from sunlight and rain, and so were much easier to carry compared to umbrellas or parasols, which had existed for nearly two thousand years by the start of the 20th century. Umbrellas had evolved to become very lightweight, but they were still too large to continue, which was a hassle when the most used form of transport was walking. Pocket umbrellas were not even invented until later in the 1920s, and many consumer stores had already started to offer umbrellas for rent when customers were suddenly caught in a downpour.
Hats had been worn for thousands of years already by the time that the Model T had been introduced, but the fashions and fashion changed over the years. Various hats signaled distinct statuses and riches, and everyone wore them from the Chinese to the Arabs, the French to the Nigerians, and a lot more. So, how can the Model T suddenly wipe them out of everyday style?