Hi,
Most of these power supplies use a switching regulator controller chip. They usually have under and over voltage functions as well as the nominal voltage regulation function. The voltage regulation affects all of the outputs including +5v an +12v outputs. Lowering one means lowering both of those.
To make the +12v output adjustable you'd have to find out what chip the power supply uses and find the voltage set resistors and change one or more, using a pot somewhere also so you can adjust the output. You might get 2v to 12v output or maybe less like 1v to 12v but it's hard to say for sure what the low end would go down to.
But you'd also have to find the under voltage set resistors and vary them too to disable the under voltage function, otherwise the power supply will shut off. You also need a certain minimum load for the output or it will shut off.
So it's not the simplest thing to do. If you build your own adjustable output stage however then you have more control and you dont have to go hunting for what part they use and where the set resistors are and all that. So you would basically build a stage that simply regulates the +12v output line down to whatever you want like 1v, 5v, etc., using a pot.
The output stage can be linear if you dont need too much output power, but for higher power outputs you would probably want to use something like a buck circuit,
The linear stages are really simple to build however, so if you dont care too much about power you can certainly do that.