I was always trying to find a way to clean the coffee stains off my stainless steel mug..Hot soapy water and a scrub brush didn't work either did baking soda or anything else I tried, until I tried bleach. About

 a teaspoon or two does it. Simply and carefully maneuver the mug so the bleach is allowed to cover the complete inside. Best done over a sink in the event of a spill. The stain removal process is almost

 instant and requires no mechanical intervention (brushing)..Works the same on tea stains too..In fact I just cleaned cranberry stains from our small stainless dishes... My daughters stainless thermo coffee
 
 pot was heavily stained inside with  a thick layer, black in colour. That came off but required two doses of just swirling to get it shiny.  In all cases, a good rinse with hot water is suggester to remove any

 residing bleach.  A final note: Bleach is nasty so use just what you need.

Bleach?  Like chlorine bleach?

Bleach?  Like chlorine bleach?

Bleach?  Like chlorine bleach?

Yes.

I use oxi-clean in a solution to clean the inside of my SS coffee pot. Soak for an hour or two and it come up like new.

Bleach works good on the Tidy Whities also.

Bleach?  Like chlorine bleach?

Bleach?  Like chlorine bleach?

Yes.

The problem is that the chlorides in bleach will attack stainless steel and discolor/damage/make it rust.  I'd suggest you don't do that.

However, you can use Cafiza 2 cleaner, which is what I use to clean my espresso machine and it will not attack stainless steel or any other metal, and works pretty much the same.  Throw a tablespoon in there, splash some hot water in there, very casual brush with a dish brush and you are good to go with no fear of chloriding your mug.