The covering is not a man. The covering is cloth and or possibly long hair. But not a husband.
Although covering with a husband is supported in other scripture, but that is not something you would want to do when you pray in public
Also, if two lie down together, they will keep warm. But how can one keep warm alone? Ecclesiastes 4:11 NIV
1 corinthians 7:3 The husband should fulfill his marital duty to his wife, and likewise the wife to her husband.
In fact it is okay not to have a husband cover his wife when they pray specifically in 1 corinthians 7:5-6
Do not deprive each other except by mutual consent and for a time, so that you may devote yourselves to prayer. Then come together again so that Satan will not tempt you because of your lack of self-control. I say this as a concession, not as a command.
1. A woman should have long hair
1 Corinthians 11:13-15 NIV
Judge for yourselves: Is it proper for a woman to pray to God with her head uncovered? Does not the very nature of things teach you that if a man has long hair, it is a disgrace to him, but that if a woman has long hair, it is her glory? For long hair is given to her as a covering.
2. If her hair is not covered she should cut it off
1 Corinthians 11:4-6 NIV
Every man who prays or prophesies with his head covered dishonors his head. And every woman who prays or prophesies with her head uncovered dishonors her head—it is just as though her head were shaved. If a woman does not cover her head, she should have her hair cut off; and if it is a disgrace for a woman to have her hair cut or shaved off, she should cover her head.
If the covering only meant long hair then it could mean something similar to the following
If her hair is not long she should cut it off. So since short hair is an abomination it should be cut shorter. That does not make any sense.
The following would make more sense to me.
The woman should be covered with long hair and covered again with fabric or cloth on top of that long hair.
However an alternative translation in a NIV footnote is as follows for verses 4-7
Every man who prays or prophesies with long hair dishonors his head. And every woman who prays or prophesies with no covering (of hair) on her head dishonors her head—she is just like one of the "shorn women." If a woman has no covering, let her be for now with short hair, but since it is a disgrace for a woman to have her hair shorn or shaved, she should grow it again. A man ought not to have long hair
Some people tried to argue that prostitutes shaved their heads and Paul did not want woman to look like prostitutes. (It is interesting that a lot of feminists like to have short hair the same style attributed to ancient prostitutes)
No matter how you look at it the minimum is long hair and a cloth may also be required. If a woman has short hair she must keep it covered with a cloth until it grows long. But men find long hair attractive so it is probably better to cover the hair whenever reasonably possible so no one is tempted to sin.
Hair also can not be braided or possibly broided. You might want to look up what the word braided/broided actually means in Greek I heard someone say that it does not actually mean braided but embroidered.
I also want women to dress modestly, with decency and propriety, not with braided hair or gold or pearls or expensive clothes, 1 timothy 2:9 NIV
In like manner also, that women adorn themselves in modest apparel, with shamefacedness and sobriety; not with broided hair, or gold, or pearls, or costly array; 1 timothy 2:9 KJV
Now regarding what I currently think is the most likely answer. Will the Churches have been following a certain custom, normally a custom does not make something morally true, but it does in this case because Paul instructed people to follow the custom in this specific case.
Now if anyone is disposed to be argumentative and contentious about this, we hold to and recognize no other custom [in worship] than this, nor do the churches of God generally. 1 Corinthians 11:16 Amplified
What might be the custom can be found in one of the writings of Tertullian called in translation On the Veiling of Virgins.
“the majority of Churches keep their virgins covered.” (verse 282 under heading chapter 2)
“Still, here (as generally happens in all cases of various practice, of doubt, and of uncertainty), examination ought to have been made to see which of two so diverse customs were the more compatible with the discipline of God. And, of course, that ought to have been chosen which keeps virgins veiled, as being known to God alone; who (besides that glory must be sought from God, not from men” (verse 283 under heading chapter 2)
The Ante-Nicene Fathers; translations of the writings of the Fathers down to A. D. 325. Alexander Roberts and James Donaldson, editors.
Or at the following website
http://www.ccel.org/fathers.html
http://www.ccel.org/ccel/schaff/anf04.iii.iv.ii.html
Anybody feel free to correct me if I made some mistakes and or typos.