White King is "mainly" STPP, not sure of the percentage. WK is the only phosphate based packaged water softner left in the US now that Amway removed "Smashing White" and Calgon switched over to sodium carbonate (powder) and sodium citrate (liquid) bases.
IIRC phosphates do not raise the pH of water that much even though they are alkaline. Certianly no where as much as say either borax or washing soda.
As for adding packaged water softener to the rinse, we are not speaking of adding heaps of the stuff. Depending upon how hard one's water is, any where from a tablespoon (or less) to 1/4 cup would do. This would also depend upon the volume of water as well, with toploading machines on high water level needing more than a modern water stingy front loader.
Many rinse additives such as fabric conditioners,softeners and starches contain some sort of water softener. Most laundry starches for instance contain borax which helps with ironing besides softening water (if used in the rinse bath as opposed to spray).
Though used mainly in commercial laundries, there are final rinse additives meant to counter act hard water minerals, especially water high in "rust" or iron, which can leave laundry brown. These additives contian some sort of "water softener" designed to sequester or some how hold said minerals away from fabrics in the rinse water.
L.