Let's look at the text.
1 Corinthians 11:1-16
"Be imitators of me, just as I also am of Christ. Now I praise you because you remember me in everything and hold firmly to the traditions, just as I delivered them to you. But I want you to understand that Christ is the head of every man, and the man is the head of a woman, and God is the head of Christ. Every man who has something on his head while praying or prophesying disgraces his head. But every woman who has her head uncovered while praying or prophesying disgraces her head, for she is one and the same as the woman whose head is shaved. For if a woman does not cover her head, let her also have her hair cut off; but if it is disgraceful for a woman to have her hair cut off or her head shaved, let her cover her head. For a man ought not to have his head covered, since he is the image and glory of God; but the woman is the glory of man. For man does not originate from woman, but woman from man; for indeed man was not created for the woman's sake, but woman for the man's sake. Therefore the woman ought to have a symbol of authority on her head, because of the angels. However, in the Lord, neither is woman independent of man, nor is man independent of woman. For as the woman originates from the man, so also the man has his birth through the woman; and all things originate from God. Judge for 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. But if one is inclined to be contentious, we have no other practice, nor have the churches of God."
You'll note there are two specified conditions for the coverings:
prayer and prophesyiing! Paul makes no mentions of head coverings while singing, or teaching, or preaching, or raising hands, or going to the bathroom or entering the service, or leaving the service, or
anything other then the two specified conditions of prayer and prophesying.
Note Paul says, "
...let her also have her hair cut off..." if she doesn't cover her head. Does Paul literally want to shave women's heads? Not likely. This is rhetoric. This is literary use of hyperbole. It is highly doubtful a woman who was prophesying without a hat was dragged out of the service to have her head shaved. There's no record of such an occurrence and such a practice would most certainly have led to an early end to the spread of the gospel.
In what context are a man and a woman not independent of one another? Scripture gives us two conditions: 1) the image of God, and 2) marriage. In all other ways men and women live independently, even in the first century. So which is the more likely context for what Paul is writing? Could be either because elsewhere Paul writes explicitly about men and women within the context of marriage but marriage exists within the context of the image of God (Genesis 1:27).
Soteriologically speaking, there is neither male nor female in Christ (Galatians 3:28).
Are we to think all women are "head" over all women? I can assure every single man here if he comes to my house and attempts to "head" himself over my wife we'll consider it mercy if he's able to walk out of my house under his own faculties. Nowhere does scripture assert such a hierarchy. So why do we impose such a measure on this passage and ignore its stated contexts?
One last clue: Corinth was a pagan city and it housed temples of Apollos, Poseidon, and most germane to this passage,
the temple of Aphrodite. This last temple is important because the Aphrodite cult was a woman-led cult. It didn't have priests, it had
priestesses. Females led that religion. Not only was it female-led but like many other pagan cults the worship of these gods involved temple prostitutes. As the gospel converted Aphrodite worshipers women leaders who were used to power and control were converted. This posed a problem for Christian worship for both Jewish and Gentile converts coming from patriarchal povs. We find the same problem arising in Ephesus where the temples of Dionysus and Artemis were central to that city's religious life. These cults were characterized by the institutionalization of behaviors God holds in antithesis to His standards. Orgies and gluttony were acts of worship! This is one of the reasons there were problems with the Lord's supper in Corinth: for the pagan converts this was a feast in which the goal was to gorge oneself to hedonistic and epicurean content (along with the fact the richer classes were able to make it to the Christian services earlier than the working classes).
So what Paul's writing has specified limits, is written with a certain amount of observable rhetoric, has scriptural context that runs through the whole of scripture, not a half a chapter in one book of the Bible, and has cultural context lost on the 21st century reader.