You don't need soap, water, a wash cloth, or a towel. You may not even need a spare shirt, after working all day in the heat. People might think you're fresh out of the shower.
Check after you step out of the shower for under arm odor. Hopefully clogged up sinuses won't prevent this check. You may discover the soap you used in the shower didn't do its job. If you discovered odor here, remember to get a different kind of soap for your next shower.
After an insufficient shower or anytime anywhere, when you detect under arm odor, put a small portion of Glycerin oil or wintergreen alcohol in your hand, rub your hands together and apply the solution to your under arm areas.
If the odor didn't totally disappear after one application, repeat the above procedure. One to three applications should take care of the problem.
Try to have glycerin oil or wintergreen alcohol with you at all times in the hot months. Apply a little to your under arm areas every time you detect any perspiration. You should also apply a little to the under arm areas of your shirt. Preventing odor build up in your shirt will likely keep your shirt fresh all day.
In extreme cases, to keep your shirt smelling fresh, spray a mist of vinegar onto the odored ares of your shirt. That is, if you have some wintergreen alcohol in a spray bottle or some cologne or fabric freshener to get rid of the vinegar smell after its application.
Hand sanitizer, also known as alcohol gel, is a possible substitute, but not an equal substitute for glycerin oil or wintergreen alcohol. This will also help keep you cool. You can set your air conditioner a degree or two higher and still be comfortable. Your body needs to sweat. It is a healthy body function to get rid of poisons and excessive body fluid. If you only exercise in a chilled air conditioned room, you'll lose almost zero weight and probably increase your appetite. Attention alcohol manufacturers, consider marketing wintergreen alcohol gel. Be the first and get a jump on the competition. 100% cotton fabrics have a better chance of staying fresh smelling with proper use of wintergreen alcohol, than do some other fabrics. (Some fabrics seem to enjoy making you smell bad, when the first drop of sweat appears).