Toss the stiff shirt in the sink, and fill with warm water. Add 1 cup white vinegar to the sink and swish the shirt around to coat every fiber with the smelly liquid. Let the shirt soak while you get your beauty sleep, and rinse thoroughly in the morning. Repeat the soaking as often as necessary to leave the fabric of your armpits soft and supple.
Pour some baking soda in an old bowl and drizzle in enough water to create a paste. Stir until the conglomeration reaches the consistency of toothpaste, and smear it over each of the shirt's armpits. Scrub the stiffness away with a heavy brush, and rinse to remove any sticky white residue. If the 'pits still feel stiff, recoat and scrub again, this time letting the paste sit for 24 hours before chipping away the dried baking soda crust.
Turn your washing machine to an extended wash cycle and plop the shirt inside the drum. Add a full dose of detergent to the water, and close the lid as the machine fills up. Let the washer agitate the shirt, then turn the washer off and give the shirt a final, overnight soak. Flip the switch back on and let the washer run, adding fabric softener during the rinse cycle. Transfer the clean shirt to the dryer and run on low heat to remove armpit stiffness once and for all.
Buy a laundry detergent specifically manufactured for athletic wear or for clothes that frequently come in contact with heavy perspiration and deodorant. Many of these detergents are made with special chemicals to wipe out stinky, stiff armpits.