Polyester can make you whiff a bit after just a couple of hours. Horrible stuff.
Cotton t shirt is a great idea, if you won't get too hot or are allowed.
You could try a product called Perspirex, or Driclor. It's meant to be for excessive sweating, and whilst you may not be excessively sweating, you are sweating a bit to get the smell in the meantime (which is normal obviously)
Perspirex you put on at night, on clean dry armpits, and it really stops any type of sweating. You wash it off in the morning (or just leave it on if you are me!) and you can then put your normal deodorant on over it, though you might find you don't need to.
Don't use it immediately after shaving, as it will sting. Even if you haven't shaved, it may sting a bit the first few times you use it. This is normal. Unless you get a rash or peely skin, you should be able to tolerate it. Proceed with caution if you have sensitive skin generally.
If you were to add this to your routine a few times a week, you probably will find a vast reduction in the smell. In addition, you can use a more "fragrant" deodorant with it than Mitchum is, like an Impulse body spray or something that you like the smell of, which along with the Perspirex, should leave you smelling of that and not BO/sweat.
I can't wear polyester in a close fitting top - 3 hours and I stink, doesn't matter whether I showered and used Mitchum, or even whether I am particularly hot or cold. I can only get away with it with the Perspirex the night before (or sometimes that morning instead) if I forgot trick. So generally I avoid it but I get you have no choice!
Good luck