My wife was saved in the Apostolic Assemblies sister organization in Mexico,
The Apostolic Church of the Faith in Jesus Christ. They also practice head coverings.
After my wife moved to America in 2001, she felt the liberty to study this issue out for herself instead of blindly submitting to what she had been taught. She came to the conclusion that the requiring head coverings today in the western world based on
I Corinthians 11 is a flawed argument. The veil was the universal symbol in most of the known world of Paul's day for the woman's submissive place below the man. The veil was usually long and not like the tiny, transparent veils used by many Hispanic apostolic churches today. Furthermore, women did not leave the house or appear in public without such a head covering. If we are going to be consistent, should not women practice the latter today if they are going to embrace this teaching? Nowhere in the scripture do we see God commanding women to wear head coverings and Paul never made such as claim. It was a man made custom that Paul felt was good for christian women of that day to follow.
The conclusion my wife came to is that Paul is saying that even though women had been made equal in Christ to men, that this equality was no reason to remove their head coverings when assembled with other believers. However, we do not see in scripture an example of the apostles commanding Christian women to wear veils where no such custom was already in existence prior to the gospel being preached in that land, and neither should we impose such a command without proper, scriptural authority.