what could cause the car to not start immediately after running the engine?

Hello, either the starter motor is not getting enough power to crank the engine, the starter motor needs too many amps to start the motor, or the motor is seized or binding causing the starter motor to not be able to turn the engine. First, have your battery -- even new - tested for state of charge and the ability to sustain a load. If it passes that, then move forward to ensuring all cables are securely connected and those connections are free of corrosion. If after this all seems well, then have the starter motor checked for binding and excessive amperage draw. If this tests good, put a wrench on the engine and crank it by hand to ensure it turns. Somewhere in that routine you will find the problem. Good luck. www.repairpal.com/repair-shops