Need advice on a good varmint barrel for my pistol contender

That's easy, 221, 222 or 223 for small to medium critters.

25, 6mm, 6.5 or 7mm TCUs, 6.5 JDJ and you can go up in size, and maybe distance

Others I am sure have other choices to fit the bill.

The recommendations from LouisB are good ones. For varmints up to groundhog size you could go .17 HMR or .22 Hornet as well. I have a .22 Hornet barrel on an Encore and recoil is so mild I have a 3-9x rifle scope on it and use a "Taco" grip. It groups 1" or less at 100 yd. The HMR can be just as accurate. The Hornet would be OK on fox and coyote as well.

How do the hornet and 222 do in the shorter barrels?

My Hornet barrel is 15" and my .17 HMR is a 14" by MGM so I can't speak to anything shorter. I have a friend who shoots well with his 10" Contender barrel in .222. The shorter you get the louder they generally get though.

with the 222 or 223, one can load down to 22 LR or 22 Mag velocities for plinking or tree rat removal.

First decide on the barrel length you want. That will make choosing a cartridge easier. I first had a 10 inch .221 Fireball, but it was so loud in the short barrel. I sold it and got a 10 inch .22 Hornet that I am thinking of having K'd to match my carbine barrel. A 12 or 14 inch barrel in .221 Fireball would be cool. A friend stopped using his .22-250 rifle because he had so much fun shooting his Super 14 in .223.

I have a 223 rifle so I would probably go with 222...both to use less powder and keep noise down and to keep brass separate. Will have to find a 12 or 14

I had a Super 14 in .223. Thing shot great and was fun.
But I needed something with less oomph (wanted 14" .22 magnum) and sold it.

I had a 221 fireball, 12" and a 17 AH also in a 12"
Both were quite accurate and effective on PDs