Veils were mainly used in the Ancient Near East at weddings. Given the matrimonial language in the previous and following scenes, there is no reason to say that a wedding veil is not in view here.1
The veil hides and reveals, and it is true that a partial glimpse of physical beauty is often more intriguing than seeing the whole. The view of the part arouses the desire to see the whole.2 This veil also represents the last and flimsiest divider between the man and the woman.3
1 Behold, you are beautiful, my love, behold, you are beautiful! Your eyes are doves behind your veil. Your hair is like a flock of goats leaping down the slopes of Gilead.