This is a swag just based on my (re) reading of the OP;
Some ‘hats’ as used in religious ceremonies, (example), may be more to impress by size and elaboration than anything else, like some military headgear.
Actually, this is a great example of where the biblical imperative is mixed in, or not distinct from, social and cultural norms. The short version is that if weather were the only cause, I doubt there’d be a difference between men and women, and certainly not between married and single women.
The Talmud derives the biblical imperative for married women to cover their hair from Numbers 5:18, which prescribes a specific ritual to be done to a wife who is suspected of cheating on her husband, and part of this ritual involves uncovering her hair. The Talmud concludes from this that the woman obviously had her hair covered before beginning the ritual, so there must be a requirement for married women to cover their hair.
Say what you will about whether that’s fair or logical, but don’t say it here, because this is GQ and we deal with facts here. The ritual is prescribed there, and the Talmud does infer from it that married women are required to cover their hair. On the other hand, the lack of any explicit command (“Married women! Cover your hair!”) certainly allows for the possibility that the hair-covering was an already-existing custom that may have been around for centuries (perhaps for weather-related reasons as C K Dexter Haven wrote), and then it became a Biblical requirement by effect of the ritual in Numbers. And single women got exempted from covering their hair, simply because there’s no ritual for girls who cheat on their boyfriends. (And ditto for husbands who cheat on their wives.)
That was certainly true of the papal tiara, which I don’t think has been used since before the days of John Paul I:
http://upload.wikimedia.org/wikipedia/en/9/9f/Johnxxiii-color-tiara-sm.jpg
http://biblelight.net/pius12tiara-sm.jpg
Mitres are often worn by Catholic brass. Archbishops and bishops in the Anglican tradition also wear mitres on important occasions; the designs and colors vary widely:
http://www.anglican-mainstream.net/wp-content/uploads/archbishop%20of%20canterbury(6).jpg
I believe it’s in the Mishna Berura, but I don’t have it in front of me so I can’t give you the exact reference.
Zucchetti. You ever watch some medieval fiction and see how the clergy have the top of their head shaved (tonsure)? It covers that. Wikipedia says it’s descended from the pileus. So maybe more of a case of convergent evolution?
It is said that one of the great targeted marketing moments of the 20th century was the realization that Jewish women buy more wigs than any other ethnic group.
A question and a comment:
I must have a block. I’ve read the above section umpteen times and still don’t understand what is being said (and rationalized) regarding women’s head covering.
With respect to the various hats worn by Orthodox Jews, different sects will have different standard ‘hats’ with the specifics of each depending on which shtetl, town, or district the sect originated.
For example, we have various bagel hats (shtreimel) worn especially by Jews from what was Galicia.
The argument is, covering your head is a sign of subordination. Men are superior to women, because women were made from men for men. Men pray with their head uncovered, because they were made in the image of God, so when they pray, they’re raised up on a one to one level with God. But women weren’t made in the image of God. They were made in the image of men. So, they should cover their heads to show their subordination.
That’s basically what that passage argues.
Not just marketing, but a healthcare niche too: They’re not an ethnic group, but cancer patients are buying a lot of wigs too. In other words, the ones who are losing their hair to chemo treatments.
My wife has brought at least a couple of her acquaintances to her wig lady. And they’ve mentioned that they’re heard (from their cancer patient support groups) that the Orthodox neighborhoods are the places to go. Many of the wig ladies have already increased their business, and at the same time developed special tact to deal with the unfortunate trauma that these women are suddenly going through.
FYI, the link you have on the word hats does not work. 
That’s it! So, my brother had to stay in intensive care at NYU Medical Center so we spent a lot of time in the waiting room. And a number of gentlemen hung out wearing that first kind of hat. (There was a Sabbath elevator too.) My parents and I had to look at each other, smile and whisper “It’s the hats” after George’s line about the Latvian Orthodox. 