There are also adapters for 3 or 4-pin fan headers to a 4-pin MOLEX connectors. use the adapter and then plug it into a PSU 4-pin MOLEX.
Yeah, but some PSU fans(especially older ones) are wired directly to the PSU PCB. You have to cut the wires, then solder them (or twist the wires together) to a 3 or 4 pin connector. I
always take out the PSU fans, some work pretty well, and some are ridiculously underpowered.
And some are hideously loud. Depending on its age, it may not be worth the effort. Even good fans have a finite lifetime - although if they spin, they at least go on the spares shelf. Kinda like the ol' folks home.