Shocker
Did you verify that the transformer you used is only a transformer and does not contain any rectifiers? You don't want a rectifier to shunt your generated high voltage, so it must be a true transformer with the low voltage secondary connected across the coil of the relay to give you the required step-up. (Important also is that the impedance of the transformer winding should be approximately equal to the impedance of the relay coil, otherwise the current will have a preference for which ever one has the lower impedance). (Output is on the high voltage side).
Try to find an open type relay and wind a few thousand turns of very thin wire on top of the existing coil if space allows it. This winding can then be used as the high voltage output. (Just a thought!).