Actually, to be in relay mode you want to press the Power button quickly three times once the GTM is powered on. That will stop the LED from flashing and turn it to steady on. Then quickly press it 3 more times. It will blink back 3 times, then the white LED will go out. To test that it is currently in relay mode, press the Power button once and it will reply with 3 blinks to confirm.
What relay mode does is shut down the Bluetooth to save energy. You’d also not want to just turn it on and leave it with the LED flashing as it would be if you didn’t put it in relay mode, as all that flashing continuously would also waste power.
I think you may have confused what the GTM paired to your device does with relay. Even when in paired mode, unless you’re using it for your own messages, whenever a GTM is on it is available to relay if it gets a message that needs a relay.
Sierra is pretty much correct. The relay just forwards the message, it doesn’t need to be on Plus, but it does need to be upgraded to the latest firmware IIRC so that it is capable of handling the message correctly, although I may be wrong about that and some of the earlier firmware may forward more than 3 hops (someone will correct me if I got that wrong.) The originator of the message needs to have Plus to send a message up to 6 hops, as well as some version of the 5.0 series of the App.