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Hi quiet whisper, welcome to the forum, it's great to see your excited about getting into hunting and have a great attitude in regards to quick clean kills, possums however are very tough little critters, I've had them hang dead in the tree by their tail for over 10mins, got sick of waiting and shot it's tail off to get it to drop, it was stone dead when I picked it up. I've had very little experience with air rifles, my trusty Ruger 10/22 is my go to possum, rabbit and hare shooting rifle, maybe you could try a pointed hunting specific pellet? Hopefully someone with more air rifle experience can give you some better advice.
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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.