Interesting topic...
Being real thin and muscular I used to wear tight t-shirts to work.
In the beginning, I could wear a spaghetti-strap wireless bra and get away with a loose dark shirt to hide my nips. Now that my girls have bloomed I now have to wear a cami or a tight fitting ribbed tank under my baggy work shirts now to keep from being "exposed". Actually I found the ribbed tank top to work best because it flattens out and lets the boob sag a bit and it doesn't have cups to hold or shape the boobs which actually accentuates them.
Most trans men are motivated with hiding their chest like most trans women are with tucking their package, I think, which is why they/we are willing to go all day no matter how stuffy or uncomfortable they/we are, and probably rationalize that someday the parts will be gone so putting up with the pain is mentally bearable. With trans women, we want our boobs and they aren't going away, so we typically won't tolerate the discomfort or pain of compressing them for long term.
For myself, work is the last place to come out, and it's a touchy subject with the job market we have; don't want to be jobless and trans. I figure that it's time to start drafting a coming-out letter or plans when the girls get to be real hard to hide

And if it's the nips that are mostly giving you away try tape, something gentle like painters tape, yeah on sweaty days you may have to reapply it but I have tried it a few times and it works with a tank top.
one more thing...

My ex-wife advised me to wear a form wire-bra as often as possible, even to bed, as it will help shape the fatty deposits as your boobs grow; I've noticed that mine have shaped nicely doing this. Can't say or prove that flattening and compressing new boobies is or is not a good thing- just something to think about