I tend to shoot off the bonnet or from a shooting chair in the back of my pick-up truck with foxes. My local land rarely offers safe shots much beyond around 300. Deer I also am mainly off sticks, usually a bipod, though I recently bought a set of ViperFlex quad sticks and though they are a bit more of a handful the accuracy is excellent. Hits within an inch of point of aim at 100 yards is no problem. I added the fifth leg just to see what difference it made and can hold rock steady at silly distances though have yet to use them in earnest - probably just get used for target practice with the fifth leg. I reckon with a bit of practice though they would do well on longer range foxes.
I also use the viperflex and rate them as the most stable platform I’ve had. I use the 5th leg when sat up waiting for Charlie, but not for stalking on foot. I can sit the rifoe on them and walk away I’m that confident in them.
They become less of a faff the more you use them.
To the op.
222, 223, or. 243
Never shot the 22-250 so can’t really comment but fail to see what it would bring to the table the above can’t do.
222 is a good round, but not as available as the others, check what your rfd keeps in stock, if you intend to home load for this then it’s a case of finding a rifle you want/like/get on with.
223 would be excellent, not so Sure about the drops quoted in this thread though ! but very capable out to 300, 53gr VMAX, cant go wrong, cheap, accurate, reliable, deadly, Lots of choice rifle and ammo. Would be my no1 choice.
243, Same as above, a favourite Calibre of mine, load it down for fox or up for deer. A versatile Calibre.
or get a 6.5cm if your foxes are in the next county
