Clean tight connections is the first thing to check. So, good job there.
You need to do some voltage testing or have your alternator tested. They sometimes get weak before failing completely. Basically, they sometimes lose 1 of the 3 "phases" generating power or it happens in the rectigier/regulator section.
If you have a voltmeter, check your battery first. It should be around 12.6 volts with engine nor running. If it's less than 12volts, your battery is getting old. Then, with enginevrunning, check voltage at the battery. Forvthe first couple minutes after starting, it should be close to 14.5 volts to replenish the battery from the energy used to start the engine. Then it should drop down at least to 14 volts, but anywhere between about 13.5 and 14. If it stays too high for too long, it can "cook" the battery and kill it. If it's not high enough, it can't recharge the battery. The alternator voltage must be higher than the battery voltage in order to put energy back into the battery.
@Ron D explains all this better. But do these checks and let us know what you have.