pull it apart, discharge the big caps, find out what the pwm chip is, get the chip reference pdf and adjust the voltage out to 36v on the 12v line, change the output cap on the 12v line to 50v, re-assemble
you'll also need to disable (i) short (ii) under voltage (iii) over voltage protection
and you may need to change the 3.3/5v line caps to higher voltage than their output as well
Given that you are asking the question in the first place, modifying the existing supply's pc board and design might be a bit beyond your reach. You're asking a circuit designed from the ground up to regulate at one specific voltage to now regulate at three times that voltage. While the process described above covers the basic steps, don't be surprised if the supply has an additional overvoltage protction circuit that kicks in, or breaks into oscillation due to control loop issues.
Another approach is to stay with the 12 V output, and add an external boost converter.
The 12V supply won't help you. The simplest solution is to get a 120VAC to 36 VDC converter. That will be easier to find than a 12VDC to 36VDC conveter.