Of course, it depends on how expansive your definition of “hat” is relative to “headgear”.
The list of types of hats in https://en.wikipedia.org/wiki/Hat includes some that are commonly worn due to religious tradition (e.g. kippah, mitre, turban, zucchetto).
Re: hats on men and certain religion. Men are usually required to take hats off inside Eastern Orthidox churches. Women are usually asked to wear a scarf covering their head. My source - personal experience touring monasteries and churches in Europe.
And this:
“While there is no specific dress code, the general rule is for men and women is to dress appropriately, modestly and respectfully, as before the Living God. Traditionally this has meant that women wear dresses or skirts that fall below the knee, while men wear pants. We ask that you not wear shorts, mini-skirts, tank tops, low-cut or strapless dresses (unless covered by a sweater, etc.). You may notice that some Orthodox women wear scarves on their heads (not hats), but this is not required. Men are asked not to wear head coverings (baseball caps, etc.) in the nave.”
This is from a Greek Orthodox Church.
Back in the mid 70’s, my all girl Catholic girls boarding school dropped the uniform and instead went to a dress code that said “please dress in clean comfortable clothes. If someone’s dress bothers you, please have a civil discussion with them personally. Be prepared to back up your opinion with facts.” In the next three years the only case of dress discussion I can remember was when we got a new (male) principal. He was fairly short, rotund and loved to wear a pink striped polyester suit with a light blue shirt and white tie, belt and shoes. We all told him that it made him look like a teddy bear (I remember numerous photographs of pink teddy bears being displayed) and he shouldn’t wear it out in public. At school was fine, but he shouldn’t wear it off campus. He stopped wearing it out.
Another thing is that the once strict observance of taking one’s hat off indoors across US society decades ago may have likely dropped due to hats suddenly falling out of fashion among large groups of men right after WWII when it was once considered almost a mandatory part of one’s dailywear.
This could clearly be seen in pictures which showed most men wearing hats in many settings before the late '40s and after.
One attributed reason for hats suddenly falling out of fashion among males after WWII was that many young men who were conscripted to fight in WWII felt hats reminded them too much of the rigid stifling military environment and its regulations and once discharged, wanted to leave as many vestiges of that environment as far behind them as possible.