Hi Sydney,
Personally, I have started covering my head, for many reasons, not just one verse.
1. The scripture you mentioned. 1 Corinthians 11:10 seems to be saying the covering displays her power/authority to speak before the messengers.
2. In the Old Testament, if a woman was suspected by her husband to be guilty of adultery, her hair was to be unbound/uncovered, this was a symbol of being humbled, also a shame. It was a shame to uncover the head, and it was also a sign of mourning. Men would humble themselves when they would pray, by removing their covering, while married women were to keep theirs on. I am not sure if an unmarried woman is to cover her head, as it is usually a sign of her husband's covering/protection being over her. Notice 1 Corinthians 11 compares a woman praying in public without a head covering to one whose head is shaved, and it's a shame. Sinead O'Connor can be thanked for bringing back the bald head temple prostitute look. Even the Temple prostitutes covered their head, though, according to the Bible, they wore a full veil.
3. When I wear a head covering, men don't "check me out". It's like I have, "Do not touch, do not covet, not available, not interested" written on my forehead. Of course, not wearing seductive clothing or makeup also has something to do with it. So, modesty. I am a curvy redhead, so if I paint my face and wear tight clothes, and wear my hair like most do these days, I look very much like a prostitute. I used to be a different person and attracted the wrong kind of people just by my appearance. I would rather not draw attention to those features the world finds attractive, when the world says to flaunt those things.
4. My husband did not ask me to do it, but after wearing it for several months, he did say he understood why I was doing it, that my motives were good, for YaH and my husband, and he thanked me for wearing it.
5.My children prefer I wear a covering. Many get offended if you don't wear one because of their own convictions.
6. You don't have to worry about hairstyles/cuts, dying your hair, thinning hair, or bad hair days when you cover your head. Our beauty should come from within, not outward appearance and hair fashions. 1 Peter 3:3
6. Last but not least, I get really nasty looks and comments from some people for wearing it, much like those nice comments from those disgusted by plural wives, so it shows that it's truth. Many men like to show their wives off and let other men drool over them and covet them and it puffs them up to have their trophy wife on their arm. It makes them feel like the big man in the room to have the woman all the other men want. One of these men once said to me with an ugly sneer on his face, "Don't people mistake you for a MUSLIM when you wear that THING on your head?" He said that in front of his wife and others, intending to embarrass me and shame me hoping it would pressure me to stop wearing it and discourage others from starting to. He didn't want his wife who looks and dresses like a worldly teenager to get any ideas. There was nothing about my head covering that looked Muslim. I just smiled and told him that lots of faiths wear these "things" on their heads, including Christians, Messianics, and Jews. He was Messianic"ish", maybe more in word than in deed. Wearing a head covering is definitely for those who don't care what others think.