It's amazing how much personality variations can affect your fencing needs. We had a buck that would push down anything not electrified, and he liked to snap electric wires and break them. He pushed gates until he broke latches, broke wood, whatever.
Now, I have a much calmer buck. My does used to like to squeeze through stuff, too, but my current does (a larger breed) aren't as sneaky that way. So we do OK with 4' wire fencing.
I'd agree that two strands, one on top and one lower, standing out from the existing fence, should do OK. Good luck. We had a rough go when our animals weren't securely fenced, and I understand the anxiousness. Panels are effective, too, but they need to be really well secured.