Hey - welcome, and you've come to the right place.
The biggest culprit in this kind of issue stems from the power running into the car to the amp gauge and back out, of which the starter relay is a big part in the chain. This happens through two connections within the bulkhead connector - the big square box that all the wires go into and come out of, located on the firewall below the regulator. Look for the big fat red wire, and the big fat black wire. Those are your in an out big power wires. All power in the car runs through those, through the amp gauge first before being distributed to the rest of the vehicle (for the most part).
It's a very bad design, and is a weak point. The design is already at its electrical power handling limit when clean and new, and add 60 years into the mix and it's falling down on the job. Typically the connections get corroded, loose, and intermittent, and sadly, is a fire waiting to happen. Especially when people add high power stereos and high output alternators to cope with the added draw.
However, we have the solution!
Search for BULKHEAD BYPASS on this forum, and talk to
@cbarge - he's tagged here and will chime in. He's the recognized authority here on making our car's electrical system safer, and in addition to the bypass, will have info on relay systems to take the load off the headlight switch, and other safety upgrades.
Let's see pics of your car please!
