In the past, maybe. I don't think it does so much now (at least in my part of the US). If it signifies anything besides aesthetics, it's signifying that the person is part of some defined group. Status doesn't enter into it.
> while disporting the mark of your superiority.
So, keeping your "low status" hat on would be OK? (I'm assuming that there is such a thing -- that some hats are "high status" implies that there exists the opposite).
Baring your head is the sign of humility. It doesn't matter which hat you have to doff.
In the UK, a "low status" hat would have been be a flat-hat, or flat cap. It used to be a worker's hat; it was also worn by countrymen - farmers, country landowners, gamekeepers and the like - but not in town.
Nowadays I often see expensive tweed flat-hats on the heads of fashionable, well-to-do townies. To me, that look is poncey.
A hat signifies status. Consider the top hat - it signifies wealth. Or the general's hat - think of Wellington and Napoleon. They signify authority.
It's inappropriate to walk into someone's parlour as a guest, while disporting the mark of your superiority. Baring your head signifies humility. It's just polite.