I don't wear any undershirt anymore. Used to wear white tees but it's too hot to double up. I've switched to a more "natural" deodorant brand called Native.
I've found that the slimmer the shirt especially under your pits, the more likely you are to sweat. Trimming pit hair helps. But looser and more breathable fabric is best. I used to sweat a ton in the Banana Republic slimfit dress shirts. I think they were partially poly material or something, whereas my current OCBD type shirts are looser fitting and cotton