All veils hook on the back. in the hair. I think I’ve seen even chassidish kallahs from rabbinish families wearing regular veils on the back. There is sometimes a style of not wearing a veil, but I’m not think it’s very common. This is worn during the whole chassanah.
The covering on the front is only from the badeken (covering the kallah – if you look carefully you can see the work “dek” in the middle, which is where the work “dek-tuch” that I mentioned earlier comes from) during the chuppah. There are opinions that once the chosson has identified her as the correct kallah and has covered her face, it is not appropriate for other men to look upon the face of the kallah during the chuppah. Nobody needs that piece of material to go down much past the chin. That’s why, LB, you saw the kind only to the chin. That’s not the “veil” that everybody thinks of. It is just supposed to cover her face. If it’s the really thick kind (like I had at my chuppah and my daughter had) you really cannot see out, but it is good to at least be able to look down underneath it and see the ground where you are going to step.
Unfortunately, amongst the more modern, they simply take the veil piece from the back and cover the face. It is very transparent and everybody can see her. I can actually tell you that it is nice to have that bit of privacy to be daavening during the chuppah. Less chance of looking around at who made it to the chuppah, etc. That would be very distracting, IMHO.