PA-28 Pilot reported alternator issues led to low voltage inflight with radio failure and possible ADS-B failure.

Narrative

The previous afternoon; the aircraft had been worked on by an A&P and returned to service for a fault in the electrical charging system. Upon startup the next morning at ZZZ1; the alternator failed to come online as evidenced by the ammeter reading zero. After a brief phone call to the mechanic; I was advised to try briefly cycling the master switch which produced no results. The mechanic said he would drive to the airport to look at the problem.In the meantime; I read through the POH which; being from 1964; is not very detailed. It does however describe resetting the over-voltage relay by turning off the master switch for a full 30 seconds. I elected to shut the plane down by pulling the mixture and turned off the master as well. After more than 30 seconds; I restarted the aircraft and the alternator came online normally.To verify everything was working; I ran the engine up to about 1700 rpm for 5-10 seconds and also switched the avionics on/off 2 or 3 times. Everything reacted normally with the ammeter showing more load as devices were turned on. I proceeded with the 20-minute flight to my destination (ZZZ) where I was to meet another person for a local flight.About 2 miles from my destination (in Class E airspace); the transmit on the radio produced static; and I suspected low voltage. I was able to land normally. I sat on the ramp for about 20-25 minutes with the engine at fast-idle and all avionics turned off; in the hopes of reducing the load so that the battery would better charge.After turning the avionics back on; we departed normally and flew for about an hour with things working as expected. When returning to ZZZ to drop off my passenger; about 2 miles from the airport; the electrical system failed due to what seemed like low voltage. We used a handheld radio to enter the pattern while making radio calls and landed normally. We were the only aircraft using the airport at the time.While we had the avionics on during flight; there is a chance that due to the possibly low voltage; our ADS-B signal became inoperative. After landing back at ZZZ; we called a mechanic to come service the airplane with a fresh battery so we could have ample electrical power for the short flight back to our home base of ZZZ1 where the charging system could be examined.There is no good way to know if our ADS-B was fully operational near the end of each flight as the voltage appeared to drop to a low state. The electrically powered fuel gauges and electric fuel pump worked normally at all times as they require only small amounts of power.