Actually, yes. A standard wash cycle with detergent followed by a wash cycle without detergent results in much cleaner clothing. It also seems to decrease the wear on that clothing.
I stumbled upon this decades ago when looking for the most efficient way of truly rinsing laundry detergent out of my clothing. The additional detergent-less wash cycle not only removed residual laundry detergent, but it had unexpected benefits.
Of course, "water Nazis" will pop their gourd over this practice.
You know what, I often run the “extra rinse cycle” because in the last decade or so, I’ve noticed that soap remains in some fabrics after one rinse. There is no such thing as a water “nazi”. There’s just capitalism: you pay for what you use. Unfortunately, the American mind is now rife with guilt over anything wonderful like clean clothes, garbage dumps, 8-cylinder engines, plastic bags, thus anything labor saving that makes life happier.