Fox are much smaller and less hardy than coyotes. Under 50 yards give or take a few, and, with very good shot placement always, it is do-able. Remember that furbearers have soft feet and don't cut the ground with hooves like a deer does. Tracking in much of the common ground clutter like grass, leaves, ect. is nigh on impossible. Also, furbearers have... uh well... fur. That is thick hair, and, sometimes longish hair that soaks up blood leaving little sign on the ground. And lastly, the little rimfire isn't going to punch much of a hole in the entrance, or, cause a exit wound to leak blood on the off side. There won't be a lot of sign to help you locate the critter after it runs out of sight. And run out of sight they will, unless you brain shoot them or break the spine. There just isn't enough energy to anchor larger animals down like a centerfire cartridge can. So, the odds are stacked against you when hunting with a Long Rifle on bigger critters.
Now, the .22 Magnum has just enough more "pop" to make it a considerably better cartridge for predator hunting. With good shot placement (notice how that keeps coming up?) the rimfire "magnum" can be effective even on coyotes to 80+ yards.
Are ya stuck with only a .22LR, is that it?