How can I get rid of dark, set-in stains in tee-shirts?

Unfortunately, I sweat pretty profusely in the arm pit area. This means that a lot of my shirts have icky stains. I never seem to see them until they’ve gotten so bad that washing alone does not work. I’ve tried Shout but that’s more of a stop-gap measure than a permanent solution. So, experienced laundry doers, is there anything I could try that would give me more than a day’s worth of stain-free-ness?

Expense is not too much of a problem. Some of these are shirts I will never be able to find again and I would hate to lose them to pit stains.