The villain is almost always slag, that gets down into the spout, and keeps the rod from stopping the hole completely. It has to come out.
You can poke at it from the bottom with a 90 degree angle scribe, or any other wire you can bend to let you get up into the spout and knock loose whatever is keeping the rod from seating properly, or simply drain the lead out, and clean both the rod, and the spout from top and bottom. I almost always found the slag on a top edge of the spout, half on and half in the hole. As much as we stir and flux the lead, Its a real PITA to try to figure out where that slag came from.
There should aways be a metal plate under the pot, and it helps to have a peanut Butter jar lid, or some other metal lid to catch the drips. They cool instantly because they are so small, so throwing them back in the pot doesn't involve much work. Just grab the lid with pliers, and dumps the drips into the pot.