I don't know. When I was a kid in the 90ies, it was still common to see older men drive a car wearing a hat. On the other hand, if your car is really not high enough to wear a hat, is it really that inconvenient to take it off when you enter? Men also stopped wearing hats in countries where owning a car was the exception. This theory also implies that somehow, before the car, mobility with a hat on was universally possible. Try riding a bike with fedora. Or a horse.

My father used to curse men who drove while wearing a hat. The hat restricts your field of vision.

I feel like cramming yourself into a subway car with a fedora is equally unpleasant. At least in an automobile you can just put the hat in the passenger's seat or something.

Worth noting that in the 1940s cars still had enough headroom to let you keep your hat on (eg. https://static.messynessychic.com/wp-content/uploads/2019/06...), but people started getting taller and cars shorter in the subsequent decades.