Will not start if engine is warm?

This is typical of a failing crankshaft position sensor or camshaft position sensor on any car brand. They commonly fail by becoming heat-sensitive, then they work again after cooling down for about an hour. While driving, natural air flow keeps them cool. When a hot engine is stopped, that engine heat migrates up to the sensors, causing one to fail.

The place to start is by reading and recording the diagnostic fault codes. That's easier to do yourself on Chryslers than on any other brand. Cycle the ignition switch from "off" to "run" three times within five seconds without cranking the engine, leave it in "run", then watch the odometer readout to see the fault codes. You can go here:

https://www.2carpros.com/trouble_codes/obd2/p0300

to see the definitions, or I can interpret them for you. There's two things to be aware of with fault codes. The first is it can take some time for the computer to detect a missing signal from the crankshaft position sensor or the camshaft position sensor,. That may not occur just from cranking the engine. Often it takes longer to detect the defect, as in when a stalled engine is coasting to a stop while you're driving. For that reason, never assume a sensor is okay just because there's no related fault code set.

Second, fault codes never tell us to replace a part or that one is bad. They only tell us which circuit needs further diagnosis. When you do see a sensor or some other part referenced in a fault code, that part is the cause of that code only about half of the time. First we have to look for other things like cut or bare wires, corroded connector terminals, and things like that.

When there are no fault codes set to direct you to the circuit with the problem, you'll need a scanner to see if either sensors' signal is missing. I have a Chrysler DRB3 for all of my vehicles. Most of the better aftermarket scanners will do the same things. Mine lists each sensor with a "No" or "Present" during cranking to show if each sensor's signal is showing up at the computer. Given your dandy description of the symptoms, this is much more likely to be caused by a sensor itself, and not a wiring problem.

This first drawing is for the cam sensor. Here's the instruction for replacing it:

The Camshaft Position Sensor (CMP) on the 3.7L V-6 engine is bolted to the front/top of the right cylinder head.

1. Disconnect electrical connector at CMP sensor.
2. Remove sensor mounting bolt.
3. Carefully twist sensor from cylinder head.
4. Check condition of sensor O-ring.

INSTALLATION
The Camshaft Position Sensor (CMP) on the 3.7L V-6 engine is bolted to the front/top of the right cylinder head.

1. Clean out machined hole in cylinder head.
2. Apply a small amount of engine oil to sensor O-ring.
3. Install sensor into cylinder head with a slight rocking and twisting action.

CAUTION: Before tightening sensor mounting bolt, be sure sensor is completely flush to cylinder head. If sensor is not flush, damage to sensor mounting tang may result.

4. Install mounting bolt and tighten. Refer to torque specifications.
5. Connect electrical connector to sensor.

The second drawing is for the crankshaft position sensor. I guessed at a 3.7L engine.

The Crankshaft Position (CKP) sensor is mounted into the right rear side of the cylinder block. It is positioned and bolted into a machined hole.
1. Raise vehicle.
2. Disconnect sensor electrical connector.
3. Remove sensor mounting bolt.
4. Carefully twist sensor from cylinder block.
5. Check condition of sensor O-ring.

INSTALLATION
1. Clean out machined hole in engine block.
2. Apply a small amount of engine oil to sensor O-ring.
3. Install sensor into engine block with a slight rocking and twisting action.

CAUTION: Before tightening sensor mounting bolt, be sure sensor is completely flush to cylinder block. If sensor is not flush, damage to sensor mounting tang may result.

4. Install mounting bolt and tighten to 28 N-m (21 ft. lbs.) torque.
5. Connect electrical connector to sensor.
6. Lower vehicle.