+1, also as an alternative to wake on LAN, you could use some kind of hardware method.

The power button to start the PC can be removed/rerouted and hooked to an external device as well, like an Arduino or another computer with GPIO.

I had a computer with a broken power button and to start it I had to touch the two wires from the motherboard power switch connector together to simulate pressing the button. An external device could simulate pressing the power button and turn the computer on from being completely off rather than in a sleep state.