All,
Many thanks for all these excellent responses.
I checked this morning, it's 3 grounding rods outside.

The older, existing, grounding wire to the well casing is definitely less than #6. It's armored cable, and WITH the armor it looks like it's as thick as a #6 conductor (without sheathing). Perhaps that's just the type of wire used to ground stuff in the 1950s? Anyhow, it's connected to the well casing with what looks like a bolt clamped onto the casing's top flange. As far as I can tell, the other end of this original grounding wire is still connected to the panel.

The underlying reason I ask all this is because I'm about to install a water softener that would electrically insulate the section of copper pipe where this grounding wire is connected to, making it irrelevant. Perhaps I should move this grounding wire downstream of the water softener? Or again, jump across it, AND across the pump to the well casing, such that all of my plumbing is grounded. Thoughts? More importantly, any reason why I shouldn't?