13 volts will not charge a modern battery, it needs 14.2 or more. Worth checking the voltage at the alternator output as it could be bad connections dropping the volts. If it's low at the alternator then it needs an overhaul, new diode pack, regulator and brushes or complete replacement.I think I'd more likely be checking for a drain somewhere. If it wasn't charging, you'd be having problems even on the days you use it back to back.
Alternator is an expensive fix for it to turn out to be something else.