Let me start out with a disclaimer: I have never shot any kind of reactive target, critter or not, with CCI Stingers at any range beyond about 40yds.
Stingers use lighter-than-normal 32 grain bullets and rely on violent fragmentation to do their job. I've shot them into water jugs and the recovered particles look like they had been shredded with a cheese grater.
However, fragmentation requires high velocity, and lighter bullets have less momentum, so their velocity typically decays faster as range increases. This is why military ammo intended for long-range fire typically uses heavier projectiles; they may start out slower and drop more, but they maintain
velocity longer, and therefore hit harder at long range.
Perhaps the Stingers lost enough velocity on their way to the skunk that they didn't have enough "oomph" to fragment when they arrived. If they didn't fragment, you're better off shooting the skunk with normal high-velocity 40gr solids or 36gr hollow points, because they'll penetrate better. (The 36gr HPs may not expand, but they may still perform better than the Stingers.)
Also, I agree with 22-rimfire; try Velocitors or Aguila Interceptors. They use standard 40gr bullets that will hit harder at long range.