My 2 year old whirlpool dryer has a 3 prong cord. My new house has a 4 prong. The dryer doesn't have a spot for 4 wires. The ground wire coming from the dryer is attached to the dryer itself. A new 4 pronged dryer cord has white, red, black, and green wires. Maybe the green wire hooks to the ground screw with the one coming from the dryer?
If you can remove the existing grounding wire from the grounding screw, then do that and connect your 4-prong's grounding wire to the screw.
I initially misunderstood your explanation of the existing grounding wire. If this is a wire coming from inside the dryer, then you'll likely need to connect it to the neutral lug once you've connected to the grounding screw a suitable ground from your new 4 prong cable. As said in a comment, your best bet is to consult the dryer's manual to verify this is the correct action for this step.