We buried the wire around our run to keep predators out. Because our lot is sloped too, we bought 4' rolls of wire and installed the wire from top to bottom - if that makes any sense to you. I don't have a good photo to show you and it's too hot to go out and take one right now, so I'll just have to try to explain it.
The top of our run is flat and covered with wire fencing. The shortest corner of the run is about 5'10" and the tallest corner close to 7'. We set the 4"x4" posts in concrete at 8' intervals and 2"x4"s in between at 4' intervals (everything on center) to create the frame for 4' wide fence panels of varying heights. We attached each section of wire fencing starting at the top of the fence panel using a pneumatic stapler. I'm not a tool person, but these were some big galvanized u-shaped staples and they're holding up great. When we got to the bottom, we cut the wire with about 2' of "extra" and then buried it. So on one end of the fence panel there's about 24" buried and on the other maybe only 20" below ground.
If you've already put up your fence, this probably won't help you, but if you've got some left to install, you might give it a try. You could also just bend the wire at the bottom to make the "apron" that some people use around their coops to protect against predators that want to dig their way in. Digging it in was the better option for us, but if it's not for you, the apron might really help.
If you do decide to do it this way, make sure you don't have the posts too far apart. It's easier to overlap the wire than it is to have to bridge a gap. It's a "measure twice" kind of thing. I'll include a picture that sort of shows how our run fence looks. If you need better ones, PM me and I'll make my way out into the inferno and take some for you. My poor chickens are eating chilled melon and grapes today because it's so dang hot. They're tough little ladies, but I've gotta show them some love every now and then. I've got a couple of peeps in with their mom and you should have seen them picking the seeds out of that honeydew. Only two weeks old, but they know what's good....
Good luck!