All of a sudden it came to me.
Remember that story about women in Bet Shemesh going for the burkha fashion style? Completely covering the head with layers of scarves?
Well, I was trying to come up with a traditional source to refute their claim of extreme modesty. It has been surmised that Leah wore a veil but that was on her wedding night (Genesis 29:25) which would only be natural although it isn't specifically stated. Rivkah uses a veil but only when she is informed that the man she sees from afar is her intended ("And she took her veil, and covered herself" - Genesis 24:65), an act which is related to marriage. In Shir HaShirim 5:7, a veil is used but again, it is connected to being in a love relationship ("The watchmen that went about the city found me, they smote me, they wounded me; the keepers of the walls took away my veil from me.").
On the less savory side, Tamar uses a veil but Yehudah presumes that because of that, she's a prostitute! ("When Tamar was told, Your father-in-law is on his way to Timnah...she took off her widow's clothes, covered herself with a veil to disguise herself...When Judah saw her, he thought she was a prostitute, for she had covered her face." - Genesis 38:13-15).
And then it came to me. Channah goes into the Tabernacle courtyard at Shiloh (my home town) to pray for a son and Eli, the High Priest observes her: ("And it came to pass, as she prayed long before the LORD, that Eli watched her mouth. Now Hannah spoke in her heart; only her lips moved, but her voice could not be heard; therefore, Eli thought she was drunk." - I Samuel 1:12-13). Do you see it?
If her face would have been covered with a veil, Eli couldn't have seen her lips. Ergo, she wasn't wearing a veil, even while in one of the most holy of places, doing a most sacred act: praying.
So, can we assume that veils - and not burkhas - are appropriate to identify a woman's marital status but when engaged in other mundane or even sacred acts, the veil is not considered obligatory?