I have also lost a scarf ring before, it was the square shape one, and then when i told my SA she told me I was using the ring the wrong way, as I would fold it into a strip, wrap around my neck and then put both ends through the ring together.
She told me the right way to use the ring, to make it more secure, is to put the two ends into the ring in opposite directions, so that after you are finished, the ends hanging will be in the shape of an "X".
Not sure if my description makes sense to you, but hope it will help.