I have a .22 air rifle which has accounted for many a possum in the willows at our place.
You'll want to get a few different pellets and see which are consistently accurate in your gun. I have decent luck with the JSB brand but your mileage may vary.
Get some pellets that are a bit heavier, 18-25 grain, the light ones are useless for tree bears.
I find the best option with the air rifle is to aim for the head and keep shooting, if it gets hung up on the tail shoot it in the head again, it makes them spasm and fall out if you're lucky.
I always move closer as I go so when they fall you can finish them off quickly if need be.
Good luck and with the air rifle practice makes perfect regarding accuracy, you'll find the way you breathe and hold the gun is very important to be consistent.
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