Chickenwire (which btw is not particularly predator-safe...) can take a good bit of stretching, especially diagonally; is it possible that it was just installed a bit loosely?
For a 5' span though it'll be awful hard (maybe impossible) to get it totally tight, and stay that way, without a top and bottom rail to staple the chickenwire to. Try using 2x4s, although they're not ideal on T-posts. This iwll also prevent you from having to diagonally brace the corners which you'd otherwise have to do (if you were just going to try to stretch it real tight).
Most predators that dig, can dig more than a foot. (they can rip apart most chickenwire too, though, so I'm not sure how much it's worth putting a whole lot of effort into burying things deep). You could consider a 2-4' horizontal 'apron' of wire on the ground, outside of the base of the run fence, either buried under turf or mulch or whatever or weighted down with rocks, pavers, concrete rubble, whatever.
Good luck,
Pat