I think your piece is missing a little flat springy piece that would be used to hook onto the edge of the pocket which you stored your pocket watch. Overalls (bibs) usually have a special place for that hook on the chest pockets.
I believe the correct term for this watch attachment is a chatelaine.
Search the Net with the term "chatelaine" and "watch" and you'll see that's what you call it.
You could also attach watch fobs to this chain as well.
Yours is nice and old.