Justin...
With You already having one wildcat to your aresnal and it being necked down to . 22 caliber whay then add a second one?. True,the . 22-250 is one of the pioneers in varmint hunting and they are very reasonable to reload but they lack distance in my book. Other than that I do not use it when I go north or west for chucks or dogs. I have a pair of calibers I like much better for long range varmints and they are my 6. 71 Phantom(its a . 264 Win Mag done by Lazzeroni) and a 6mm Ackley Improved(which I had built locally). I do realize the Lazzeroni is a tad heavy or more than what you want,but do not overlook the . 264 Win Mags as they are one of the best double duty calibers out there.
Now other options to consider would be these and I shoot both of them at times also,the . 243 Winchester and the nearly forgotten . 25-06. The . 243 will kill almost anything you want to shoot. I have taken all kinds of game with this gun and it is a pleasure to shoot. Varmint loads are fast and accurate and with a nice selection of 85-100 grain heavy loads most any game you lay in the cross hairs will drop. The . 25-06 is a gun you will find if you ever get you hands on one is one gun you'll not want to do without. Its as flat shooting as most any of the newer wildcats and very reasonable to reload.
The . 220 Swift,although a very fast cartridge,is one I own and do not care for. The caliber is very hard on barrels and is well known for shooting out a barrel anywhere from 400-800 rounds thru it. Its not nearly as accurate beyond 300 yards and on those high heat,heavy humidity days,is well known for air bursts of the bullets due to its high velocities. I am a Pre-64 Winchester Model 70 collector and if it wouldn't be for rarity of this caliber in one of these guns,I wouldn't own one. Needless to say,if this gun is needed to shoot then the . 22-250 or the . 243 goes out instead.