This maybe a silly question.

Was the 97 Elantra GLS equipped with something to lift the rear suspension? Perhaps an electric motor of some sort?

I ask because when I shut my car off I noticed a tick/thumping sound coming from the rear.

I was starting to think, maybe it was the rear end lowering, perhaps the motor, letting it drop.

Please advise. Perhaps it's a rear suspension issue.

 

Sorry, there is no stock motor lifting the rear suspension. They use to make an aftermarket air-shock that had an inflatable blatter that could be used to lift the rear end higher off the ground, but those either inflated manually or came with a small compressor that buzzed to control pressure. I doubt anyone would bother installing such a device on an Elantra.

The sound is most likely related to the electric fuel pump mounted inside the tank underneath the back seat of the car or a valve closing/opening on the vapor emissions return circuit. There is a small cover under the back seat you can remove to expose the top of the tank and provide access to the pump location. Try removing the cover and see if the noise can be isolated while someone else turns off the car.