I wear a chapel veil (In Spanish, mantilla, French: mantille). People tend to use scripture and the writings of various Saints (St. Paul, St. Augustine, etc), or cite modesty as a justification.
The reason I wear one is more accessible. Our altars have a covering, and so do tabernacles. Sometimes, as at Saint Peter's basilica, the entire altar is under a canopy. The chalice is covered during parts of the sacrifice...
I think of my head as the residence of my intellect, and I want it to be almost ritualistically veiled, so that God can reside their, or be placed their, in some sanctified way. The Blessed Virgin is a good example of this...
In addition, my I grow up with my family members wearing veils for cultural reasons, even at NO masses; my mother only stopped wearing hers as I got older, because people made her feel ashamed with their stares and whispers 
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My understanding is that they should be worn when you're within a building where the tabernacle is, particularly in its immediate vicinity, whether or not there's a mass.
