Any one of a variety of symbolic coverings of persons or sacred objects as a sign of reverence,
Then we also have the humeral veil, the chalice veil, the ciborium veil, a bridal veil, at Lent we veil the crucifix…
I think the word veil is used because it is seen as a reverent type of headcovering. I also think that most would interpret St. Paul’s admonition to cover our heads to mean that we are to cover our hair since that is what he is particlarly mentions. A hat in most cases can not cover the hair - a veil or mantilla (which is the same thing) would be necessary.
newadvent.org defines 1 Corinthians 11:10 Therefore ought the woman to have a power over her head, because of the angels. (Ideo debet mulier potestatem habere supra caput propter angelos) in this way A power… that is, a veil or covering, as a sign that she is under the power of her husband: and this, the apostle adds, because of the angels, who are present in the assemblies of the faithful.