Stopped while running

Was it running before you worked on it? If so, double check your work.
Make sure battery has a full charge and battery connections all good. Did you check good voltage and ground for fuel pump motor at the wiring connector?
If it cranks good but won't start, have a helper crank it, while you check for spark at the spark plugs. If spark everywhere, use a gage and check proper fuel pressure, I don't care if the pump is new, check fuel pressure, anyway. Also, check fuel injector pulse. Only crank the starter motor a few seconds at a time, let it cool off.
Are there any applicable trouble codes? How long since over-all tune-up?

I'm going to give you a link to a dudes youtube videos. Before you throw parts at it you need to learn to test ! You need to find out what is missing , fuel , spark , compression ! this guy is a teacher at an automotive trade school and has work for 20 years as a automotive technician ! How to test fuel pressure injector pulse and spark with basic tools He has hundreds of videos !

When vehicle reaches a high temperature and cuts out pore water on the crank sensor. Once crank sensor is cooled down , vehicle will be able to start.

Have you checked the timing belt, it is most likely has stripped out teeth and not turning the camshaft(s). Pull the belt cover off the front of the engine and see if the camshaft pulleys rotate when cranking the engine over.

See if your idle air control valve needs to be cleaned out. This is a common problem in cars that stall out a lot.

distributor,or crank sensor

Did you check your fan clutch? If it runs hot running around town, but cools going down the road this could be it. also check radiator for obstructions like bugs or bent fins.

DEFECTIVE CRANK OR CAM SENSOR.

Step 1. Find a OBDII scanner (OBD stands for On-Borard-Diagnostics). Find out what DTC code (or codes) are stored in the ECU (DTC = Diagnostic Trouble Code, and ECU = Engine Control Unit - it's the computer which manages engine operation).
Step 2. After you find out which code is stored in the ECU, go to http://www.hmaservice.com/ and register your veh (by VIN). Afterward, you can navigate the Hyundai Service Site to find out what system is giving you problems, the diagnosis procedure and what may be needed to fix it.
Nuff said... Let me know what you find.

I had a similar problem and had an engine tune up done and the car runs almost like new now. Mine is a 2001 Hyundai Accent too.