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I'm assuming no codes are setting, which will definitely make it harder to diagnose. It may require someone diagnosing it at the time it's acting up.
If it only happens on a hot restart then it may be fuel related, and that's nor something that would normally set a code. When you shut the engine off after driving the high underhood heat can cause the fuel in the rail to boil and pressure to elevate. Fuel can then push past the check valve in the fuel pump and push fuel out of the line and rails. This leaves mostly vapor and little fuel in the line so it causes a very hard start. Replacing the pump would be necessary to repair this.
It may also be an ignition issue, but ignition would normally set a code. It's not unheard of though for a cam or crank sensor to internally short and take down both sensors, which will prevent it from setting a code.
If the problem only seems to happen on a hot restart that's significant and something you'll want to mention to a shop so they can try to duplicate it under the conditions where it happens.
Sounds logical. Yes, no codes were displayed. Pep Boys said they did a fuel pressure test, & it checked out. We'll take it to someone else that is more familiar with Saturns.