Trouble is, I can't get that last bit of solder that capillaries in between the leg of the IC and the wall of the thru-hole.

I use soldering aids to try to break any remaining solder so the lead is free of the plated through hole. If I can't break it free, I apply more solder and try the sucker again. For cases where I can't make progress, I heat the joint and wiggle the lead away from the hole.
If it's the last pin on that "side", I'll use an iron to melt the solder and pull it out from the component side. If I have a pin on both sides, I work on one side and then the other.