I've got a couple of .22's, and one bullet I've enjoyed using (and I've forgotten the name - it's been a while) was a long-range cartridge, with a somewhat hollow-point bullet. They were designed to have more power, too, and packed a bit more grain.

I sighted in the scope so well on one of my rifles that I could put these precisely through a critter's eye and then blow apart the opposite side of its head. Gosh, it's remarkably less gruesome when you're actually doing it and not typing it. Hmph. But rest assured, these animals were all evil.

I've shot the pump-action rifles you speak of, and they're a lot of fun. Peep sight?