Waking up a host in standby on any network traffic is extremely likely to wake it up right after it went to sleep - keep in mind that lots of packets travel your network at any time, even if you think it's idle. E.g. any ARP request from any other host is sent as a broadcast, so it'll wake up the host.
A sleeping host cannot distinguish this "useless" traffic from "useful" traffic (in the sense of your intention).
Wake on LAN requires a specially crafted "magic packet" that targets a device's MAC address. This implies someone else specifically sending this special wakeup call.
This could be done with a service running somewhere else on the network either detecting an external wakeup event or listening in on the network traffic (e.g. ARP for the server's IP address) and then sending a WoL packet.
If you've got no suitable device for this service there are many cheap single-board computers around that could do it. The server in question would need to support proper WoL from soft off or hibernation, of course.