There is a well known problem with staining with selenium toner. I know that Ansel Adams recommended a process similar to what you are suggesting, but my experience has been that I get serious staining if I put a print into the toner without first having processed it for several minutes in hypoclear.
I suggest modifying your after-session process as follows:
Refix in fresh fixer
Rinse in plain water
Hypoclear for a minimum of three minutes. Do not discard the hypoclear.
Back to the hypoclear for another minute. Discard after this step.
And I would wash the prints by soaking them in trays of plain water. Each soak would be at least 5 minutes, with the prints transferred one-at a-time to another tray of water, and then repeating this process 5-6 times.