Yeah you would need to unwind it, the wire is soldered to the magnet.
I take it you have never seen someone make a pickup since you're asking this question.
They take the magnet, solder the wire to it, put the casing around it, then the wire is placed by spinning on a machine that counts how many turns it's made so you can accurately make consistent sounding pickups.
Pickups take thousands of turns of wire, you cannot undo the wire, swap the magnet and then re-wrap it in wire. Well you could, if you were a pickup maker.
