I think you are doing the right thing to set up a visit with your orthodontist
. Having a part of your braces
fall off is not an emergency, as it will not affect your the treatment of your teeth immediately. However, obviously its needs to be corrected and the other braces checked as well to make sure they are properly attached. Calling your orthodontist's office and describing what happened should be sufficient, as I am certain that they will give you the first non emergency visit possible to have the problem fixed.
In the meantime, I would be more concern about the potential for trauma to the gums or mouth. I say this because sometimes when a part of the braces falls off or breaks there can remain a sharp edge or piece of wire that can poke the gums or inside of the lips or cheek and cause pain and bleeding
. If you have this problem, then that obviously changes this a bit and you should ask your orthodontist to be seen more quickly and have the problem fixed.
As always, feel free to call your orthodontist with any problem related to the care or maintenance of your braces.