Must a woman's head covering in church hide all her hair, and can she just wear a hat?

In the history of the Catholic Church, from the 1920s through the 50s, most of the time women wore hats, not veils, because hats were more fashionable, though you would see a mix among younger children at First Communion. All Paul talks about is a covering for your head; you put on something that is obviously covering your head, and it does not have to cover every hair. When Paul says a woman's hair is a covering, he is not saying that is the covering he requires; there is another covering she is to wear to show that she is under the authority of her husband. Why? Because, as Paul gives the reason in 1 Timothy 2:11-15, Adam was formed first, that is the order of God's creation, and Eve was the one deceived and who led the human race into sin. The covering is her acknowledgment in her heart that she accepts this order, so that she will live under it without giving trouble to the church and her husband. When I was in first grade, for a woman to enter Mass without a covering was treated like a mortal sin; girls would scramble for a tissue and a bobby pin if they forgot their hat. The reason so many women stopped is that no one teaches them the reason, and the 1976 pontifical biblical commission treated it as a mere passing custom, which does not surprise me, because they are liberal; but this is not what the Church used to believe.