I have a nail/staple puller from the old days. It has a slide hammer and all that, but I never use it. It does not work that good.
I have moved quite a bit of fencing for my sheep, but found the easiest way to pull the wire from the staples is to actually use a generator, a hand held grinder, and a thin metal cut off wheel. Using that, and (5) cut off wheels, I successfully removed, and then reinstalled, 10 acres of sheep
fence. That does not sound like much, but it was around half a mile of wire, staples multiple times to posts 16 feet on center.
Note: You would think when moving
fence you could just pull the wire up, fence posts and all, then pound them back into the ground, but it dos not work that way; it is impossible to get the fence tight. Instead, the wire must be removed from the fence posts before being it put back up in the new field. The grinder helps cut the staples off in quick fashion.