I have a major boggle. I use a gel stick antiperspirant/deoderant. Everytime I put it on I get a stain from the product. Isn’t it supposed to guard against the stain? Am I putting too much on? I feel like a fool, do I NOT KNOW HOW TO USE DEODERANT GEL STICKS? I’m a grown man for chrissakes!!!
So! Use a different deodorant, maybe? 
I’ve found the gel ones to be less likely to stain, and just as effective.
Are you getting “pit stains” from sweat, or are you getting stains from the product itself?
Perhaps you need to switch brands. I’ve found that the gel kinds I’ve tried don’t work as well as the solid ones for me, and they run out faster to boot. YMMV, of course.
My husband used to get stains from his (white solid, not gel) deodorant/anti-perspirant on the bottom of his T-shirts all the time as he put his shirts on. He won’t switch types of deodorant because he is very particular about scent and effectiveness in this area. His solution was to start rolling his T-shirts up from the bottom before he put them on. That way, even if the deodorant did rub off as he put his shirt on, it got onto the inside of the shirt, not the outside.