That is some mad, mad shit.
The bit that is missing for me from that article that it doesn't explain why god needs you to specifically cover your hair (surely an omnipotent being can see through cloth? Don't start me on why an all powerful being who would get so worked up about follicles could be a being that would be in any way something to worship...)
The piece goes on about women being sacred child bearers (blugh to that), but if the sacredness is in the child bearing, surely covering your tummy and your hoo-ha would square that off?? Where does your hair come into it?
From a quick bit of research, that ol' woman hater Thomas Aquinas weighs in with "the man existing under God should not have a covering over his to show he is immediately subject to God; but the woman should wear a covering to show that besides God she is naturally subject to another." Apparently it's from Corinthians (Paul was not big on women either, IIRC). Man is the glory of god and should therefore not be covered up, while woman is the glory of man and should therefore be covered up, because she's his possession and not for others to look at.
It's interesting that all of the Abrahamic religions do or did require women's heads to be covered. I am assuming that it is just garden variety misogyny - long hair in the regions that spawned the Abrahamic religions is associated with women (Paul makes reference to this as being the natural state of affairs - short hair on men, long hair on women), so it's viewed as a secondary sex characteristic. Because it's a secondary sex characteristic that belongs to women and therefore it's impure or unclean, it was written into the faith that women had to cover up.
I do wonder what mental contortions religious feminists have to put themselves through. It seems to require such a thorough cherry picking of the holy texts that I can't honestly see what the value is.