Do women have to wear head coverings all the time?
There is no verse in the Bible that suggests anything other than that a woman should never pray or prophesy with her head uncovered. It doesn't say a woman has to keep her head covered ALL the time. If she wants to, that's fine, but nowhere in the Bible will you find that women have to wear head coverings continuously.
1 Corinthians 11: 4. Every man who prays or prophesies with his head covered, dishonors his head. (Please see head coverings for men.) 5. And every woman, who prays or prophesies with her head uncovered, dishonors her head; for she is on a level with her whose head is shaven. 6. For if a woman be not covered, let her also be shorn; but if it be shameful for a woman to be shorn or shaven, let her be covered. 7. The man, indeed, should not cover his head, because he is the likeness and glory of Elohim: but the woman is the glory of the man. 8. For the man was not from the woman, but the woman from the man. 9. Neither was the man created for the woman's sake, but the woman for the man's sake. 10. For this cause ought the woman to have on her head the mark of authority, because of the Messengers. 11. Nevertheless, the man is not without the woman, nor the woman without the man, in our Master. 12. For as the woman was from the man, so the man is by the woman; and everything is from Elohim. 13. Judge for yourselves, among yourselves; is it becoming that a woman pray to Elohim with her head uncovered? 14. Does not nature teach you that in a man, if his hair grows long, it is a reproach to him? 15. But for a woman, if her hair is abundant, it is a glory to her; for her hair is given to her for a covering. 16. But if any one is contentious about these things, we on our part have no such custom, nor has the assembly of Elohim. (AENT)
 It's a shame for a man to grow long hair, if his goal is to look like a woman. Torah requires that there be a clear distinction between male and female (Deuteronomy 22:5). Each person must be clearly recognizable according to their natural birth gender. Long hair on a man is NOT the sole issue; those taking Nazarite vows wore long hair some for life (Samson) and others for a season. Paul himself took a Nazarite vow. Rav Shaul is teaching against pagan sexuality practices, of those who do things to make themselves appear as the opposite gender. This an issue of gender distinction and understanding the Image of Elohim in both male and female.
Or rather, as an additional covering, i.e., on top of proper clothing for worship and having her head covered. This does not mean a woman's hair is in place of a head covering.