Amazing that it lasted as long as it did. That was a failure just waiting to happen.
I assume by "double" piece you are thinking you can get a piece of coping long enough to replace both the loose piece and the piece that is spalling above it in the picture. That would work better than just putting it back as it was but I would worry that it was still cantilevered. It will depend on how closely you can match the existing coping, and your tolerance for the mis-match, that does look like a pretty standard coping style (nearly identical to what is on my pool, except my sections are 24" long).
If the mortar under the stones is thick enough that you can make room for a piece of metal as Allen suggests that will work as well. I recommend 316 stainless and at least 1/4" thick by 1.5" wide. onlinemetals.com is a good source for small pieces like that.
Another idea is a pair of short SS angle irons, fastened to the side edge of the existing pieces with concrete anchors such that the loose piece can then rest on the angles.