Forgive me but I need to digress to answer your query.
Since washing photographic materials works by diffusion of the "bad" stuff outward from and area of high concentration, the film or print, to an area of low concentration, the wash water, and since the speed of the migration of the fixation products is determined by the relative concentrations of byproducts in the emulsion and the wash water, I've been convinced that the strongest washing is accomplished by total changes of water. In other words, the most effective wash method is by fill and dump since you reduce the concentration of fixation byproducts in the washing environment radically every time you dump and refill.
The key for me was reading, long ago, that Kodak rated wash effectiveness by the number of complete changes of water. That got me thinking.
I can't remember where I first read that Kodak specified for washing by continuous flow, a certain number of complete changes of water, but I took that to heart and applied the concept to do a fill and dump wash sequence.
The short description of my sequence was that I filled a small plastic tub and agitated prints in it for something like a minute, dumped it completely and refilled. Then, repeat, repeat , repeat ... I believe I repeated a total of ten times.
To elaborate, the first couple of changes were quite quick, since my initial objective was mainly to rid the wash environment of the majority of fixer clinging to the surface of the print. If, by some rough guesstimate, one can reduce the surface "cling" of fix by a factor of a hundred or a thousand at each of the first two fill-and-dumps, the fix clinging to the surface has become a non-issue by then. The "cling" would have been reduced by some ten thousand fold to a million fold. What remained of fix and by-products largely remained within the emulsion. After that, I then slowed down the sequence to allow diffusion to do the heavy lifting. The rest of my cycles were at least a minute or two long. The overall idea is to have the print live in the cleanest possible environment as much of the time as possible. That gives diffusion the greatest opportunity to do it's happy work.
If I recall, Kodak specified something like ten(?) complete changes of water in one or another pamphlet. They mentioned - I think - a test using dye to see how completely the water got changed.
The fill and dump system works exceedingly well. My oldest prints, some thirty plus years old, are in very good shape despite deplorable storage. Would that my old composition and printing skills have been anywhere near as competent as my washing...
I have never since really trusted any washer that doesn't do complete dumps, so this system felt like a safe and reliable way to wash prints. Any method that just swashes contaminated wash water back and forth around one's print seems, to me, little better than washing one's photos in a toilet.
Now, to get back to your question of good management of water usage ... if one doesn't use endless gallons at each fill and dump cycle, you should get the most washing bang for your buck via fill and dump.
I used to just maintain a largish tub with tempered water in it, and I used a beaker to ladle fresh wash water from the tub into the wash tray after I had dumped it. Much faster than refilling from a faucet. And as a bonus, it is easier to temper a large volume of water just once, than to fiddle endlessly. Of course, a thermostatic valve might be a worthy alternative, but my sequence is simple, fast and solidly effective. I like simple, fast and effective.
Of course, with fiber prints, wash aids are mandatory for good washing, speed and water usage efficiency since fiber prints are the most challenging washing job.