Single Tub Wringer Washers
Before Tide P&G's most popular laundry soap was "Chipso", indeed it was one of if not the best selling laundry product for it's day only falling from grace when Tide was introduced and slowly at first but then as a stampede housewives chucked soaps for wash day.
Indeed P&G was initally worried that introducing Tide would kill off demand for Chipso, which it did but sales of the former more than compensated.
Aside from Chipso there were many brands of soap powder on the market, some names are still around today but as detergents. Fab, Duz, Rinso, Ivory Flakes (the powder came later), Lux Flakes, American Family, etc.
Ivory Snow and Flakes along with Lux were seen as "fine white laundry soaps" designed for the household's better wash such as fine linens, silks, etc. Brown bar soaps such as Fels, Kirkman's, and so forth where marketed as "heavy duty" and packed the power to deal with badly soiled laundry such as Johnny's mud caked jeans.
Anywho we digress:
When using a single wringer washer most housewives/laundry workers would ring the wash out of the machine and into either a stationary tub or sink filled with hot water for the first rinse. Some would have two tubs side by side (often with a wringer between) so to move wash from one rinsing tub to a second tub for another rinse and or to blue. This method allowed the hot soapy water in the wringer to be used for sucessive wash loads and not tie it up with having to rinse.
Depending upon how the family's finances stood rinsing was done manually and the wash either wrung by hand or second mangle/wringer was used (if the family could swing it) for rinsing. Of course depending upon how much wash there was to be done one could simply wash and mangle the lot by machine, hold it off to the side, drain the washer, refill with fresh water and start the rinsing. This would have been a very wasteful use of water to some though.
Problem is when using soap for washday you don't want that hot wrung wash sitting cooling waiting to be rinsed. If allowed to cool too long the all that soap/soil would settle down back into the fabric and that is not good.
Oh and the other reason for wringing right from the wash water is if the items were going to be boiled.
Boil washing is always done on pre-washed/soaked laundry for best results so the wash came out of the wringer, soaped and placed in the boiling pot.