It is a useless battle that some people get pulled into. People can see through your button shirt? Gasp! Put on another shirt. Oh no, people can now see your undershirt! What to do, what to do? Ok put on another shirt between your button shirt and your undershirt to prevent people from seeing what might be on your undershirt. Gasp, people can see your inbetween shirt...
I once had a roommate with that exact same logic. He ended up wearing 3-4 shirts every day. Then he had a crazy shirt washing pattern. The outer shirt is "dirty" from outside elements. The innermost shirt is "sweaty" from wearing it. So those two would need to be washed after every use. But the 1-2 inner shirts are clean and will be slowly rotated to become the innermost shirt therefore reducing the amount of washing needed when you wear 3-4 shirts at a time.
But soon it didn't end with button shirts. He had to wear undershirts and inbetween shirts for all styles of clothing. With all those layers under a T-shirt he looked like the Pillsbury dough boy (except for where his arms were visible).
Don't go down that path. I never wear anything under a button shirt. If the button shirt is so transparent that you can see yourself through it, then it is a crap button shirt and don't buy it to begin with.