Hair covered or is it rather Head covered?
1 Corinthians 11:5 KJB - But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven.
The "dishonoureth her head" is a reference to her Husband [house-band, the true wedding-band], since the "husband" is the "head" of the "wife" ["body"] and Christ Jesus the "Head" of all mankind, and the Father, the "Head" of Christ Jesus:
1 Corinthians 11:3 KJB - But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.
What is the "covering" of the woman's head then, that she have not "head uncovered", a piece of cloth? No, it is her "hair", which is her "glory":
1 Corinthians 11:15 KJB - But if a woman have long hair, it is a glory to her: for her hair is given her for a covering.
Even as the entire context refers to "hair" [and "shaven", "shorn"], and references Genesis, where there was no "cloth" on the head, but only that which God created for the woman, her "hair", which is that which is "covering" the "head":
1 Corinthians 11:7 KJB - For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man.
1 Corinthians 11:8 KJB - For the man is not of the woman; but the woman of the man.
1 Corinthians 11:9 KJB - Neither was the man created for the woman; but the woman for the man.
1 Corinthians 11:10 KJB - For this cause ought the woman to have power [authority] on her head because of the angels.
Notice this reference, of Mary, wiping the feet of Jesus, and the dust thereof, with her glory, her hair, in otherwords, she laid her glory, her pride, her self, in the dust, that she might forever serve at the feet of her Lord Jesus:
John 11:2 KJB - (It was that Mary which anointed the Lord with ointment, and wiped his feet with her hair, whose brother Lazarus was sick.)
Again, the context speaks of the "hair" and length thereof, not cloth:
1 Corinthians 11:14 KJB - Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him?
This was also to the way in which man was created from the beginning, and also a reference to the Nazarite vow, since all those who took the Nazarite, were to be unshaven, unshorn, and to go about with long hair, not to their glory, but to their humilty [for they were symbolic of Christ Jesus's humility]:
Numbers 6:5 KJB - All the days of the vow of his separation there shall no razor come upon his head: until the days be fulfilled, in the which he separateth himself unto the LORD, he shall be holy, and shall let the locks of the hair of his head grow.
Numbers 6:18 KJB - And the Nazarite shall shave the head of his separation at the door of the tabernacle of the congregation, and shall take the hair of the head of his separation, and put it in the fire which is under the sacrifice of the peace offerings.
Numbers 6:19 KJB - And the priest shall take the sodden shoulder of the ram, and one unleavened cake out of the basket, and one unleavened wafer, and shall put them upon the hands of the Nazarite, after the hair of his separation is shaven:
Judges 13:5 KJB - For, lo, thou shalt conceive, and bear a son; and no razor shall come on his head: for the child shall be a Nazarite unto God from the womb: and he shall begin to deliver Israel out of the hand of the Philistines.
Judges 16:17 KJB - That he told her all his heart, and said unto her, There hath not come a razor upon mine head; for I have been a Nazarite unto God from my mother's womb: if I be shaven, then my strength will go from me, and I shall become weak, and be like any other man.
Finally, Pauls' dealing with the believers, who argued over such trivial matter as head coverings of women [between Jew and Gentile believers, etc], Paul took them back to Genesis as the foundation, and then says of local 'custom':
1 Corinthians 11:16 KJB - But if any man seem to be contentious, we have no such custom, neither the churches of God.
There are other great references in scripture [KJB] on this subject, which help add to the depth which is herein given, such as in Isaiah, Jeremiah, Ezekiel, Revelation, etc..