You can always just do the paperclip trick which is pretty much free. Rarely do people use 2 PSUs these days however back in 2013-2014 it was the norm pretty much. All you need to do is short 2 pins on the second PSU and as soon as you push the switch it will turn on.

A few safety things.

Make sure the PSUs are grounded together somehow.
Don't mix +12V between the 2 PSUs. Use a volt-meter to test and confirm.
On the aux PSU, hook up some fan so in the rare case that the main PSU shuts down, the aux psu won't be running without a load

If you are good with electronics you can hook up a relay so both PSU start and shutdown at the same time.