Whether a woman should cover her head in public or not has often been a topic of debate in the church; the practice was evidently common in ancient societies, and is still observed in some more traditional cultures. God does address the topic in Scripture, but not so clearly that may be confidently dogmatic about it, hence the debate.
As a guiding principle, we should observe that God has defined sin in His Law, and He has explicitly commanded us not to add to it (De 4:2); so, we shouldn’t expect God to authorize anyone in the New Testament to change the definition of sin by adding new commands; only to provide commentary, to help us see more clearly what He intended from the beginning.
So, when God focuses on head coverings, we should note that He doesn’t point us back to any specific command in Torah; He opens by stating the roles of men and women in relation to Himself and each other: Christ is the head of every man; and the head of the woman is the man. (1Co 11:3)
This evidently implies that a man who prays or prophesies with his head covered dishonors his head (4); it also implies that when a woman prays or prophesies with her head uncovered she dishonors her head. (5a) What exactly is implied here, why is it implied, and how do we know this?
What we feel when a woman is made bald (involuntarily) tells us all we need to know here (5a) — there’s something unnatural about it. When it’s considered a shame for a woman to be bald, or to have all her hair shaved off (6), in any way that’s different than for a man, we’re admitting we know this principle organically.
We know from the Creation story (Ge 1:27) that men should not cover their heads: they’re made in God’s direct image (7a); women should cover their heads because they’re made in the image of man, or indirectly in God’s image. (7b) The Creation account further reinforces this concept in the fact that Man was not made from Woman, but Woman was made from Man (8), and since Woman was made expressly for Man, and not vice versa. (9)
In other words, the details of Creation imply that  women have a different role in the home and society than men do,  that this role is a submissive, supportive, helpful one (Ge 2:18), and  that it’s appropriate for women to reflect this role difference symbolically by covering their heads in public. (10)
This does not mean men are better than, superior to, or more valuable than women; their mutual interdependency proves this. (11-12)
So there’s a natural law, evident to all in tune with Creation: women should cover themselves in public (13), and men should not. For example, nature itself teaches us that when a man covers his head with long hair he brings shame on himself. (14) However, the opposite is true for a woman; when she covers herself with long hair it’s a glory to her; her hair is given to her as a natural covering. (15)
How we maintain our hair is thus the primary way we reflect God’s design here, and the biblical text plainly states this — so, technically, this has nothing to do with a material covering over the hair. Mandating that women cover their hair with a material covering at any time doesn’t stand up to scripture: it isn’t commanded in Torah, it isn’t clearly inferred here, and it isn’t explicitly discussed anywhere else in Scripture.
Further, as far as the biblical standard is concerned, how long men and women should generally try to keep their hair, or what style they should use to represent this role difference before God, is evidently cultural since no particular hair length or style is specified in scripture. Apart from those who are overtly defying, blending or reversing biblical roles, what do people in any given culture perceive to be a natural or appropriate hair length and style for each sex? This is the only guide we have, and it’s subjective and relative for a reason; as it is with the types of clothes which further distinguish the sexes (De 22:5), this is determined by culture.
All this said, it’s clear that we might adorn ourselves with headwear in ways which accentuate our appearance. In doing so we ought to apply the same principle; we differentiate ourselves in our respective roles by how we treat the place in our bodies where we consciously reside, in our minds or heads. Ornamental or symbolic coverings should evidently be treated much like hair; for men, not covering more of the scalp and neck than where the hair is naturally growing as a covering of the skin, and larger, longer coverings for women.
For example, the priestly headgear required by God in the service of the tabernacle seems to follow this pattern: the mitre, crown (Ex 29:16) and bonnets (28:40a) weren’t shameful for the men to wear, but were glorious and beautiful. (40b) They didn’t violate this principle even in spirit; the style evidently covered little more than where hair naturally covers the scalp.
So then, when men wear a hood to keep from freezing in the bitter cold, or from burning in the sun, do they violate this principle in spirit? Since this type of clothing isn’t symbolic but more practical — and the context here is clearly symbolic — I think it’s safe to say that this isn’t a violation of God’s pattern in any sense.
And finally, what about the Nazarites, men who didn’t cut their hair for long periods; some never felt a razor their whole lives. (Jdg 13:5) No problem: long hair can be wrapped up in a turban and kept off the neck and shoulders in public – it need not be a shameful covering simply because it’s long.
In matters which are not clearly specified in scripture, let’s study it out for ourselves until we’re convinced of what’s right for us (Ro 14:5), being careful to follow God’s precepts as best we can without rigidly imposing new regulations on others, but each pursuing alignment with God according to our own conscience. (Ro 14:22)