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If you or any animal are not grounded when touching the wire, you will not receive a shock.
That being the case, you can run a second wire near the first one, on top of the fence. This second wire needs to come from the ground rod near the box.
Keep them a few inches apart, two inches minimum. Ideally, you want the critters to touch both wires at the same time when climbing or sitting on the fence.
One thing to keep in mind when setting the distance between the two wires is the possibility of arcing. You definitely don't want the wires to arc to each other, which can happen if they are too close to each other. You may want to check for that before settling on a gap between the two wires.
Definitely keep the other ground wire connected to the pole. You would have to lay the wires on the floor close together, and plug the unit in from a safe distance away, to check for the arcing. I doubt it will jump a 2" gap, but it wouldn't hurt to confirm that. Also, keep in mind that this increases the odds of a person being shocked if they touch the top of the fence.
Are you still with me? Do you need further assistance with this question? If not, I'll be turning in for the night.
No, that would not produce a shock. It requires touching the hot wire and the ground wire, or the hot wire and earth ground (soil).