hi Mark,
The enamel covering on the old wire will be dried out and brittle, if you unwind and rewind.
Unwind the coil, count the turns and note the winding direction.
Get some new enamelled wire of the same gauge.
Make a mandrel using a bolt and two pieces of plastic.
Fix the bobbin on the mandrel, place the mandrel in a handrill chuck, clamp the drill in a bench vice.
Wind on the same number of turns in the same direction.
You must fill each layer of turns if you have a hope of getting the same number of turns on the bobbin,,, dont pile wind.
Being mains, I would add a little extra insulation on the lead out wire from inner layer.
Good Luck.!