Overchlorination doesn't lead to the smell, but it can dry out your skin.

Where I live, free chlorine concentrations up to 10.0 ppm are legal, but pools are normally set closer to 2.5 or 3 ppm. Even at 10, you can't smell a chlorine smell, but my skin won't feel great after.

However, chloramine concentrations as low as 0.2 ppm can give a chlorine smell.

One of the ways to get rid of chloramines is adding a lot more free chlorine, enough to break down the chloramines.