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.

That's been discussed here before. I was kind of pissed on for agreeing with it. After most of the smoke was over and the party disbanded, someone (thaumaturgy, iirc) pointed out that cars became more aerodynamic because of the higher speeds of highways, and that might be the missing link as to why interstates and cars helped kill men's hats. Before that, cars were boxy enough to still wear a hat in one.

Edit: Having now read the high heel article, they aren't 100% correct about heels being useless in the muddy streets of the era. Staying up out of the mud was part of the point of heeled boots of the three musketeers type era.