The thing is, it doesn’t have to be a mantilla in the Latin church either.
I just don’t know how the mantilla became de rigueur for traditional Latin Catholic women.
The mantilla is Spanish so I suppose it’s possible that Mexicana/Philipina women wore them a lot in those days, but in the parishes I frequented all my life (I worshipped all over Chicagoland, USA) they were not to be seen until the SSPX came to town, and then usually on the younger women who perhaps had little to no memory of how it was for us before.
I am getting on in years, but when I was growing up all the women covered their heads, and no one wore mantillas. At least not in my area. The women wore scarves of a variety of colors (but very simple), hats and a popular doily kind of thing they pinned on. My grandmother, a Latin rite Roman Catholic from Poland, always wore a babushka (the word means grandmother, but we also used it to mean the scarf), which was a scarf no different from what the Ukrainian Catholic women were wearing. And she did not take it off after Mass, she wore it outside in the garden and shopping on the street.