Once a big enough blob gets sucked up into the shoulder, it's pretty much over. You done messed up, already, and you might as well bring out the solderwick.
The trick is to not get that much solder on the iron/pad/pin to begin with. Large bevel tips are forgiving of the amount of solder on there. You do all the work with the flat cut face in this type of application, and gravity holds the solder on this cut face. You can get a large area of the tip in contact with the board (and hence the flux, which after it melts is very "shallow") in order to keep things working, properly. And when the tip is running lean on solder, the bevel tip is excellent at sucking out small bridges. The bridges in the toes, what sometimes can otherwise get left behind between the last two or 3 pins of a drag solder operation, not the big blobs that suck up into the shoulders.
Liquid or paste flux applied to the board is a must.
There are some SMD parts that are more sensitive than others. Yeah, sometimes you need some super fine solder wire. And sometimes, you might even look around the board for a reliable source of just a tiny bit of solder. Touching the iron to a nearby other pad on the fluxed area of board might do the trick.