If they are barbed wire fences I untie (using pliers) the middle strands from a couple of adjacent posts, spread them apart, walk the goats thru, and retie. It's a hassle, but the alternative is to go along the fence until you find a gate, which can be a long ways.
In the goat 1st aid kit are a pair of pliers and a small roll of galvanized fence tie wire. If the original ties break or are old and weak I replace them. I like to leave the fences in as good of shape, or better, than I found them.
Fences are one of the best places to find sheds. It seems like when they jump over and land on the other side it jars them enough to fall off (same with crossing draws). My vote is following the fence to a gate and hunting along the way. That will also give you a good idea of where the more regularly used crossings are, and you can search out from those. That way, if you find a big one along the fence, there is a good chance of finding the other nearby. Of course this is all speculation like most of horn hunting. My experience is mostly mule deer winter range out in the sage, so I don't know if any of those theories apply to what you are doing, and I don't know if you will have as many gates. Also, if there are two-tracks along the fence, you risk losing out to road hunters. I am skilled at the art of glassing for horns from a truck/ATV, but I am still a firm believer that traveling on foot is a huge advantage. I plan on using my goats for it, and my dog is trained to find them. Like a good friend of mine says, there's always another horn out there.