If you're going to limit yourself to a 10" tube, I'd look in the .221 Fireball or smaller category. The .222 Remington would be OK, too. The .223 Remington works in a 10" barrel, but it really shines in a 14" barrel.
In a 10" barrel, I'd look at .221 Fireball, .22 Hornet, .22 K-Hornet, .218 Bee, .22 Jet, or maybe a .222 Remington. The .221 Fireball and .222 Remington will be pretty close to each other in a 10" tube, and both will be way ahead of the Hornets and Jet with the Bee being somewhere in between.
If you were to step up to a 14" barrel, the .223 Remington, .219 Zipper, and .219 Donaldson Wasp are all good candidates. The .226 JDJ would also be worth looking at. But all of those would be extremely loud out of a 10" barrel, and also less efficient.