The remote regulator Delco charging systems were mediocre at best when they were new, and 50-something years of age hasn't made any improvements. The first thing to do is to get a factory wiring diagram and make absolutely sure the circuits from the alternator to the regulator and under-dash wiring is all intact and does not have any loose connections, corrosion, etc. No shortcuts here- - - -you've already tried the shotgun parts at it approach, which didn't help. The hum noise is probably coming from the field relay which is supposed to activate the alternator on startup. It's one of the two mechanical contacts inside the regulator cover.
There are solid state regultors available that eliminate the mechanical points and keep the appearance of the original equipment regulator. If you have some compelling reason not to retro-fit the system to the 10-SI internal regulated alternator, at least go to the electronic remote regulator. They're available on Ebay, and they're cheap! Be sure to wring out the wiring first!
Jerry
