There is more than one cause of staining:
1) Insufficient fixing leaves some silver salts in the emulsion, selenium toner will 'tone' these salts to a reddish color. The stain is usually blotchy, staining the parts of the print where the fixer didn't get to because of poor agitation, print floating on the surface or prints sticking together.
2) Expired fixer produces an overall stain because there is silver salt left everywhere in the emulsion.
3) Prints taken from heavily used fix straight to the toner can develop a blotchy overall stain because the carried over fixer contains a lot of dissolved silver, which the selenium again gleefully tones.
4) Acid fix can cause staining in heavily used Se toner where prints are transferred from an acid fix with insufficient washing and carryover has used up the pH buffer in the toner. More often, the carryover results in the precipitation of black gunk in the bottle of working toner solution. Kodak, an advocate of acid fixers, used to recommend diluting the toner in HCA. This practice would have increased the buffering capacity of the toner but the resulting mixture would last only as long as it took to oxidize the sulfite in the HCA, usually a matter of hours. The technique was only useful in large commercial operations where the toner would have expired by the time the HCA oxidized and the resulting toner mix would be disposed of. Kodak has removed this recommendation as it doesn't work in amateur darkrooms.
I think the only guarantee of no-stain toning is a 3rd fix in fresh fixer (which is rotated through to the 2nd fix, which becomes the first fix, which is disposed of) followed by a rinse, HCA and a good rinse/wash.
A light alkaline bath before the toner may be extra insurance but the carried over alkali will eventually, again, use up the buffering capacity in the toner. I don't know the effect of extra alkali in Se toner, but it shouldn't be hard to find out - put a teaspoon of borax or carbonate in the toner and see what happens. Sulfite may be a better choice as an alkalizing bath as it would oxidize in the used toner to neutral sulfate.