For brimmed hats, the problem in cars (at least in my experience) is not crown-to-roof clearance, but the fact that the brim hits the headrest, forcing you to lean forward uncomfortably (or to squish the brim).
Looking through car advertisements, it seems like headrests became common in the mid-to-late Sixties, after hat-wearing had already started to decline.
Cars didn't always have headrests. They are much more common (maybe even mandatory) is now for safety reasons. Just look at old cars from the 60 and you can see they didn't have headrests.
Yeah, that's why I mentioned that headrests didn't become common until the late Sixties. My overall point was that maybe cars were _not_ the cause of the hat-wearing decline.
Looking through car advertisements, it seems like headrests became common in the mid-to-late Sixties, after hat-wearing had already started to decline.