It’s a tradition of the Church (lower case t as they say

) and is a sacramental. Other sacramentals are holy water, and the rosary. Up until 1983 it was mandatory that women cover their heads at Mass (see 1 Corinthians), and now it is just optional. Just like not all people say the Rosary, so not all women cover their heads. (hats work too - same idea)
The idea is that in the Catholic Church’s tradition and all throughout the OT, things that are veiled are holy… like today we veil the Tabernacle. The reason that women are holy is that God gave us a very special thing, that we are able to be co-creators with Him. We are the very last thing He ever created, and inside of woman God can create a new soul. And because what is sacred is veiled, so women are too

It’s a lovely compliment to women, and yet a bunch of idiotic feminists called it suppression. Well they definitely did suppress the beauty of femininity, and through their efforts and others’, head coverings have fallen out of use. It is interesting to note that protocol still dictates that women veil thir heads when in the presence of the Holy Father (an audience). But how much more important is our Eucharistic Lord than His Pope!!
The reason it’s more prevalent in the TLM is the same as why incense is more prevalent, why Gregorian chant, the music of the Church, is more prevalent, why the language is more prevalent. We want to be as Catholic as can be, and we accept ALLLLL the teachings of the Church, even the little things like women veiling their heads.
Btw, anything would do, hats, scarves, anything. Lace is just a bit of a trend in the area of head coverings at Mass now, and many will say that hats were a bigger thing in the 50s and such. It doesn’t matter the difference, when your head is covered, it’s covered

Some people try to argue that chapel veils specifically are far superior(vs hats/scarves), but that’s their opinion, not the Church’s.