I thought I had problems with critters around where I live! I think that gators also dig. I know they can dig quite deep to bury eggs so if they wanted to get under a fence I'm sure they could. I would pretty much guess that it is not one lone gator. They don't live alone generally. They seek out other gators during mating season.
Anyway, I would contact a local fencing person who knows about how to make a fence that really keeps out gators. At least in the backyard! I would think it would have to go into the ground at least a foot.
We have a lot of wildlife here and one of the first things you learn is not to leave any kind of food outside. Not even your garbage. It needs to be kept locked up in a garage until garbage day. If you cook out take extra precautions. Animals are all about scent and long after you are done cooking they are still smelling that food!
It just draws them into your area all the more.