WARNING: NOT FOR USE IN HIGH HUMIDTY.
Adding more water to an already hot and humid situation risks a Wet Bulb.
If the water coming out of your cold tap is cool (which it should be, since pipes are typically underground), then I think maybe it would still work because you could refresh the t-shirt with colder water occasionally. It’s just less than ideal compared to doing it in a dry climate.
Fun fact: the reason air conditioners are called “conditioners” instead of “coolers” is that they were originally designed for dehumidification.