Ideally, I see nothing wrong with a woman, whenever she ought to pray, covering her head with a head covering. It was popular in Ireland before the Council, along with some other Eastern European Catholic countries.
1 Corinthians 11 is speaking about the need for women to wear veils out of submission to their husband/father (whoever is their superior).
Here is the passage before verse four:
 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.  Every man praying or prophesying with his head covered, disgraceth his head.
Also, 1 Corinthians 11:7-10:
 The man indeed ought not to cover his head, because he is the image and glory of God; but the woman is the glory of the man.  For the man is not of the woman, but the woman of the man.  For the man was not created for the woman, but the woman for the man.  Therefore ought the woman to have a power over her head, because of the angels.
1 Corinthians 11 is stressing the importance of females veiling as a sign of submission to their head.
St. Paul purposefully uses the term "head" while describing the male-female relationship because the "head" refers to both the real human head and the head of the created hierarchy. Covering the female head is the recognition of their place before men, as being constructed of man rather than being built in the image of God. Bishops, Priests, and Monks visibly submit to God with their dress
(cassocks, habits, etc.) and do not cover their head in submission to other men (the opposite of why women are commanded to cover their head, and why men aren't commanded to cover their head). Clerics and lay brothers cover their head solely out of reverence and adoration for Our Lord (women can do this too, but it's more precise to say they veil out of submission).