From your description, it makes sense that you would jumper +5V AUX (green) to PSON (white). Note that if pin 16 was labeled PSON#, that would imply you need to pull it low to activate (inverted logic).
It appears that +5V AUX is at +5V while the power supply is off and for 2 seconds after turning on.
Instead of jumpers, try using just 2 diodes as follows:
[color=green]+5V AUX [/color] ---|>|--- PSON
[color=red] +5V[/color] ---|>|--- PSON
Hopefully this will keep the power supply on.