Only one way to know for sure. Get out your pocket knife and do an autopsy. There are "head shots" then there are "head shots". Do a little skinning and find out what your pellets did, where they hit and how they performed.
I've killed enough possums to recognize a good wound track versus a bad one. I got put on possum slaying duty when I got my first shotgun (an NEF in .410) at the age of 11. Both body and head shots work well with those, but body shots too close can be messy (I disemboweled one shot from about 5 or 6 feet broadside with that shotgun). With .22 LR, I go for head shots only (easy enough to make, especially with a scope), and all are one shot kills (unless I miss all together), though if they're twitchy, I'll finish them off.
With the air gun, I know my first hit was a glancing blow, no penetration, just an angry possum. The second hit the jaw bone and entered, but probably tumbled off course because of that. The third hit in the bottom of the neck and made it to the brain, but I don't think it went far enough for a humane kill. All shots were almost straight up (possum was in a tree) at no more than 10 - 15 yards (hard estimating distance upwards for me).
If they are all just under the skin or flattened out against the skull and were placed to be a "brain" shots then you've not got enough gun. Better be looking for a bigger caliber. If you shot him thru the nose, broke a jaw and exited thru the neck then you probably got enough gun, just poor shot placement. Personally, Ive always thought possum were pretty tenacious critters and not that easy to kill cleanly, even with a .22 long rifle.
I've never had a problem killing them with a .22 LR, but again, head shots. Most are 10 yards or less, just because that's how far the possums get. I've had a couple get out at 20 yards (paced off and calculated), then turn around and stare at me like they were daring me to shoot. Both were one shot kills (scoped Remington 597). I've done one raccoon at that same distance with the same gun, again, he turned and looked at me. I have no doubts about the .22 LR on possums or coons, as long as the shot placement is good.
The .22 caliber air gun, I'm not quite so sure about. I was using Daisy hobby pellets (wadcutters), which I'm sure now weren't sufficient, and the only reason I used them is because I knew where my gun would put them. I do have a couple of pellet samplers similar to what blhof suggested, so I'll have to figure out which pellet my RWS likes best before I try it again, I guess.
Short of recommending you get a big bore airgun, there is not sure way to advise you for clean reliable kills. Maybe you already have an adequate combination (maybe not) but its gonna take some skinning on your part to find out for sure.
Silvertp
If my current air gun just isn't enough, it will become a plinker and I'll stick with my .22 LR while I consider an alternate air gun. I intend to start skinning possums and raccoons soon enough anyway just to get some practice at preserving hides (I think it would be a neat skill to have). Guess I'll take the time to study wound tracks while I'm at it.
Thanks for the suggestions, guys. If anyone else has some insight, please, chime in.