Q: 2009 Hyundai Sonata tries to start while already on

My 2009 Hyundai Sonata (automatic transmision) is having issues when I start it up. Sometimes when I turn the key to the ACC or ON position the car will start up as if I turned the key all the way to start. Other times when I turn the key all the way on and let it go the starter will continue to try to start the car and I can hear the starter gears grinding. The key does not get stuck and falls back to the ON position correctly but the starter continues to stay engaged and grinds. If I wiggle the key the grinding will stop. Additionally, sometimes the car will crank fine with no grinding but when I go to put it in reverse or drive the grinding will start again.

This was happening for a few weeks and the starter finally went out. I've replaced the starter with a new starter but it is continuing to do the grinding. Im thinking it may be the ignition switch or the wiring but don't know how to detetmine which one it is. Any help is appreciated.