Woolite
I do what you do, but my damp white micro fleece cloth comes from a 2 gallon bucket filled with the hottest water I can stand and a couple of ounces of Woolite, which was a tip from a very savvy table guy on this forum.
I have a small square of Simonis 860 HR Tournament Blue cloth left over from my table install. I plan on taking it to the dry cleaners and asking them about stain removal. When I eventually recover my table with more 860 HR I will have more cloth to experiment on. I am also going to experiment with Woolite and cold water for stain removal, as I believe 860 HR is a 70/30 % blend of wool and nylon.
For some reason Simonis, who many regard as the makers of the worlds best table cloth, does not offer any tips on removing stains from their cloth, at least none I am aware of. Maybe there are just to many variables beyond their control in removing stains from their cloth and they do not want the liability issues that may result from recommendations.
I wish they would do some testing and just offer the industry standard disclaimer warning of testing it on a small piece in an inconspicuous area.