Yeah that source of chlorine often goes unnoticed but it's a fact that we inhale a vast amount of chlorine from shower vapors (lungs) and through the skin. As far as I know a shower filter is the best method to reduce chlorine and other pollutants like metals. Aquasana and Aquapure pro filters have good filtration system but im sure there are others that accomplish the same and are cheaper.
I’ve also read somewhere that adding vitamin c to bath water helps with the chlorine levels.
Mike