Alternator

There is a big variance between the quality of rebuilt alternators, it may very well be a bad alternator or the wiring or high electrical resistance somewhere in the circuit. A simple and easy to check for the alternator , Turn on the ignition switch to the on but not run position do you see an icon on the instrument panel representing a battery, if you do not see this battery icon the alternator is not seeing power to allow it to charge. Start the engine does the battery icon go out. A more complex but good diagnostic test is checking "voltage drop" on the battery ground and power cables. You can find a video on how to do a voltage drop test on youtube. Put a volt meter across the battery terminals with the engine off, at rest the battery voltage should be 12.6 volts if the battery is fully charged. Now get a friend to start the car while you watch the voltmeter, the voltage reading across the battery should read approximately 14.6 volts if the alternator is charging correctly.