Head coverings for women in the New Testament
Some see head coverings as a valid practice but make it an optional choice, some do not. There are always some groups, mostly sects and cults that want to maximize minimal passages. Some think certain dress and traditions from the law make them more spiritual, or “God approved.” Mandatory Head covering can come under this category.
This is the only place in the New Testament this head covering is mentioned and probably because of the women whose heads were shaved for their pagan religion. The Corinthian church had struggles with their false religious upbringing they came from and so the whole letter is about their disorderly conduct and divisions that were being caused contrary to what is instructed by the apostle Paul.
First, Paul is giving an ontological spiritual order. In v.3. To understand this we must read the portion of the letter through.
1 Cor. 11:2-7 “Now I praise you, brethren, that you remember me in all things and keep the traditions just as I delivered them to you. 3 But I want you to know that the head of every man is Christ, the head of woman is man, and the head of Christ is God. 4 Every man praying or prophesying, having his head covered, dishonors his head. 5 But every woman who prays or prophesies with her head uncovered dishonors her head, for that is one and the same as if her head were shaved. 6 For if a woman is not covered, let her also be shorn. But if it is shameful for a woman to be shorn or shaved, let her be covered.”
He is addressing praying and prophesying by women in the assembly. He goes on to say in verse 10: "For this reason the woman ought to have a symbol of authority on her head, because of the angels." 1 Peter 1:12 tells us that angels watch man. Therefore, a woman's head being covered shows her submission to the angels of God's authority being over her.
1 Cor. 11:13-16 “Judge among yourselves. Is it proper for a woman to pray to God with her head uncovered? Does not even nature itself teach you that if a man has long hair, it is a dishonor to him? But if a woman has long hair, it is a glory to her; for her hair is given to her for a covering.”
Paul says her own hair is a covering, he then says, v.16-18 “But if anyone seems to be contentious, we have no such custom, nor do the churches of God. Now in giving these instructions I do not praise you, since you come together not for the better but for the worse. For first of all, when you come together as a church, I hear that there are divisions among you, and in part I believe it.”
What are these divisions? Before he goes onto the next one, which is their taking of communion, he makes it plain that noone of them is to defend this matter and become contentious and disrupt their unity.
A "covering" on a woman's head is used as an illustration of Gods ordained order, headship, and the authority of God over man and man to his wife. It is when approaching God in prayer or when prophesying.
This covering is not only a cloth but a woman's hair length. The verse in context "Does not even nature itself teach you that if a man has long hair, it is a dishonor to him? But if a woman has long hair, it is a glory to her; for her hair is given to her for a covering" (1 Cor. 11:14-15).
woman whose hair is long shows she is a woman and not a man (meaning down
her back). The Apostle Paul is also saying that a wife's hair being longer
than her husband's shows her submission to her husband’s headship. Again
Paul’s context is explaining the spiritual submission in the order of God.
This like the kippah, yarmulke, a skullcap, worn by Jewish men. The origin of the yarmulke is basically unknown though some trace the practice to biblical times, when the priests in the Temple were instructed to cover their heads, but it is not in the Bible.
Some teach Jewish men to wear a kippah at all times, and especially during prayer is  According to the Rambam. Jewish law dictates that a man is required to cover his head during prayer, but there is no mention of any claim that Jewish law dictates that a Jew is required to cover his head at all times.. Others explain that it is to Thank God for "crowning Israel with splendor" (Talmud – Brachot 60b) The Talmud says that the purpose of wearing a kippah is to remind us of God "above us" (Kiddushin 31a). (Talmud and Mishnah)
Basically, it is used as a sign of respect and fear of God: acknowledging God is above him. Some use it to identify themselves, as I have seen a number of Messianic Jews wear the kippah.
So with the head covering for women, it is a personal choice and not something that should not be used to judge spirituality. Unless I missed something, Paul summarized it with this
But if a woman has long hair, it is a glory to her; for her hair is given to her for a covering. But if anyone seems to be contentious, we have no such custom, nor do the churches of God.”
(1 Cor. 11:15-16)
Paul says to the Corinthian church, we have no such custom, nor do the churches of God. The apostles did not instruct this as necessity, only for those in Corinth who head was shaved..
“[Neither the churches of God] The churches elsewhere. It is customary there for the woman to appear veiled. If at Corinth this custom is not observed, it will be a departure from what has elsewhere been regarded as proper; and will offend these churches. Even, therefore, if the reasoning is not sufficient to silence all cavils and doubts, yet the propriety of uniformity in the habits of the churches, the fear of giving offence should lead you to discountenance and disapprove the custom of your females appearing in public without their veil.”
(from Barnes notes)