Dogs can jump a 4 foot fence pretty easily. So can foxes, and raccoons will go right over it. That said, we live in an area with coyotes, dogs, bears, cougars, hawks, ospreys, eagles, wolves, etc. We had a problem with a coyote trying to get into the coop twice, our dog ran him off and he hasn't been back. The neighbor's great dane scratched around a bit and gave up and went away and hasn't been back. The chickens are left alone, even when they're free ranging.
In town your biggest concern will actually probably be the coons and neighborhood dogs. The chickens can escape a 4-5 foot fence pretty easily if they really want to, but any medium to large sized dog can get in there even easier. If you don't want to cover the run that's perfectly fine, but consider raising the fence to 8-10 feet. You can go get some 2x4s (used or new) or see if somebody can't hook you up with some long wood poles off their property or something, and just attach them to your fence then staple chicken wire to them.
I would also take an extra 2 feet of chicken wire and bury it a few inches under the ground and then let it sprawl outward, cover it all up with dirt, or bury it straight 2 feet underground to deter digging.