Soap doesn't dissolve well in hard water. Hard to rinse it off things.
Try an experiment. Buy a gallon of drinking water at the store. Wash your pan in the normal way using only this drinking water and soap in the usual way. See if the soap smell remains.
Long term, a water softener for your house will improve the washing of all things, people, dishes, clothes.
"I must say as to what I have seen of Texas it is the garden spot of the world. The best land and the best prospects for health I ever saw, and I do believe it is a fortune to any man to come here."
Davy Crockett, 1836