Another tip - use plenty of solder! Yes - the sucker works better if
there's plenty of solder on the joint. In fact, it's useless for
removing those residual bits that you're talking about. If you can't get
it in a few attempts, re-solder the joint and try again!
If you *do* have a very small bit of residual solder, try heating the
joint and then wiggling the pin with a small screwdriver as you remove
the iron and the solder solidifies. This won't allow it to re-attach the
pin to the PCB in most cases.