If your new (AC?) relay will work with half-wave rectified 24VAC, then diode OR'ing it to the other relays should work. I would try it with one diode and the relay, to see if it pulls in and drops out appropriately, first. The other possibility is use a 24VDC relay and filter capacitor...with the diode OR circuit.
Added: I just tried running a 6VAC relay (didn't have a 24VAC) with just a diode from a 6VAC transformer. It pulled in, but buzzed loudly. Added a 500µF/16V capacitor across the relay coil, and it worked perfectly. I think your idea will work.
Ken