You can absolutely do this--you don't even have to be hesitant about it. Many extreme users link PSUs to power their TECs or GPU array.
They make cables for it now, you just need to link gnd with the green (pwr on signal) and purple (standby) cables and it will start right up with your main PSU.
Honestly though, you can be much lazier if you want. I have two such systems right now (on a box on my desk xD) powered by two PSUs each for testing our company's components. How do I have them linked? I turn on the system...then jump the PSU powering our hardware...
Most expansion cards/hdds don't need power instantaneously with the motherboard.