Does the head covering need to be visible to both people and angels?

I’m not sure this view is accurate. Two problems: First, if we assume that angels can only see things on earth from a higher position (hence, they could see a head covering but not something else, such as a skirt), then we’d better have all our services in an amphitheater because otherwise how would the angels know that a woman even has a head covering on? The text simply says that women should have their heads covered ‘because of the angels.’ No one knows for sure what that means; it is certainly a dangerous thing to assume one interpretation that is both speculative and unlikely and then base one’s conduct on that.

Second, to argue that a head covering has to be distinctive so its meaning is clearly understood I think is not accurate. The reason is that head coverings were used in the ancient world by various groups. They were not distinctive of Christian women, though they were normative for Christian women. Thus, to make them distinctive today is not in line with the custom of the first century.