Q: My 20

My 2003 Saturn Ion intermittently doesn't turn on. I can drive it, turn it off, then it won't turn back on. But this doesn't happen all of the time. In the last 4 months, it has happened only 6 times. I noticed that the problem occurs after the car has been idling for a few minutes,i.e., I turn car on and make a 3-4 minute phone call, then turn the car off. Then it won't turn back on until a few hours later. The car has approximately 260,000 miles. Otherwise the car runs great. It has a brand new battery (bought two weeks ago) and the starter is only a year old. The check engine light DOES come on periodically but it recently passed a smog certification (back in October 2016).

Basically, I never know if it's going to start or not which puts me in a "stranded" position too often.

Any thoughts on what could be the problem? Someone suggested that there's a connector/coil to the battery that could be faulty.

Thanks!

My car has 260000 miles.
My car has an automatic transmission.