For this cause ought the woman to have power on her head because of the angels. The point being, in order to uncover, she must have been previously covered. This is because Sha'ul used different Greek words for the natural hair covering and the headship covering. Katakaluptos basically means to UNcover or UNveil. So far, we have a 'men uncover, women cover' command.
Hair is more like a glorious decoration given to woman.
Now if Sha'ul had meant the naturally occuring hair covering and the headship-type covering to be one and the same, he would have used the same word for each. If such a natural covering sufficed, then Sha'ul is wasting his time teaching this since the women already had a natural hair covering. Sha'ul deliberately used different words for the two coverings so we would understand they were complementary to each other but not identical. The natural order is to either wear a covering over the hair or to fully exploit the shame of being uncovered by also shaving off one's hair too. Now among those who agree a married woman should wear some sort of covering, there is always the one who argues this headcovering is due to modesty.