1999 Plymouth Neon battery light stays on

Do you think the charging system is ok? I realize the alternator has been replaced, but, sometimes the wiring to or from the alternator maybe faulty. I'd use a digital multimeter, with everything turned off, take a reading across the battery posts. Generic specs say around 12.6 volts is a fully charged battery. Start the vehicle take a second reading, the second should be higher than the first? If it isn't, take a reading at the alternator bat terminal. Is that reading higher than the first?