I've just serviced my .22 HW45 and .177 Brum tempest.
What a difference that makes! Both are much more smooth and pleasant to shoot and the Tempest is noticeably more accurate.

A simple strip, clean and re-lube the spring with moly grease and and they both shoot like new (better than new in the case of the HW45, which are usually horrible when they leave the factory).

A trick I learned a while ago, the HW45 piston chamber and piston head must be bone dry when assembled, I use brake cleaner spray to get all the old grease and oil out. The HW45 is such a beast than even a tiny spec of oil in the chamber will cause dieseling, which will make it loud, inaccurate and blow out the breech seal. BTW I once chrono'd a dieseling HW45 at well over the specified limit (it was shooting orange sparks out of the barrel, someone had sprayed WD40 into the cylinder).

In the case of the Tempest a light machine oil on the cocking link and moly grease on the sliding shoe makes it so much smoother to cock.

Maybe don't rush out and buy a tuning kit, maybe just give your springer a careful service and see how it feels?