I have the compression shirts from underworks. Binder fabric on the inside; cotton tee shirt material on the outside. This I assume so you can wear it as a shirt. I have a tank top/wife beater and a muscle shirt.
What I do is wear them inside out, with the cotton against my skin. I used to use baby powder to absorb the moisture, but stopped when I started T, as the gel is kind of messy and I have to wait for it to dry before I can get dressed, and by then I forget the powder.
Also, if I'm going to wear a button down shirt, having the shiny compression fabric on the outside makes my tee shirt slide over it and not bunch. I usually only do that if I'm going to work where it's fairly cool with the air conditioning. I doubt I'd wear that many layers outside in the heat.
I am so looking forward to having top surgery in Dec!! Counting the days!!
Jay