I don't know the composition of the Ilford toner, but I've had success with Kodak Rapid Selenium Toner (usually around 1+6) when I take the print directly from the second fixer into the toning bath. No rinse, HCA, or wash before the toner. The rapid fixer is ammonium thiosulfate and that is a major component of the KRST also. This procedure kills the toner more rapidly than using the normal rinse, HCA, wash before the toner, but I don't recall ever having stains. The procedure must lower the pH of the toner and it will eventually cause a precipitate to form, but I've also found it to enhance split-toning effects. If I were not interested in split-toning, I'd go the normal route (but eliminating the drying step after a thorough wash).
After rereading the OP, I note that rinse, HCA, and thorough wash cycles should both precede and follow the normal toning step.