Running wire for new subpanel in garage, how to best utilize existing wiring considering the scope of my project?
First, the 6/3+gnd cable can be pushed to 60A.  If it is certain types (NOT NM or UF), it can be pushed to 70A.  

You need 2 grounding rods unless your grounding rod passes a particular impedance test which requires some expensive equipment to do.  For most people a second grounding rod is cheaper.  Those are called ground rods.  The wire from the panel to the ground rods is the grounding electrode.  

You're not allowed to send 2 sets of the same kind of power to the same outbuilding. Except, your installation was approved and therefore is grandfathered. The approval was not an error because AHJs have the right to waive requirements like this. 

And #1/0 Al or #2 Cu wire is good for 125A. (technically 115A but you get to round up to the nearest breaker.  You would be better off using Aluminum wire for this type of feeder, especially this much of it at this size.  You generally go 2 sizes larger when selecting aluminum instead of copper.

Honestly given that you are grandfathered as-is, I'd be maxing out the existing (free) 6/3 w/g cables before sweating any upgrades.  There are 2 legs to this: 


Upgrade the subpanel(s).  Your first priority in panel selection is plenty of spaces. Then make sure both its busing and its main breaker are large enough for your feeder.  I recommend main-breaker panels because you must have disconnect switches, and you can't have more than 6 so you can't force people to snap off individual breakers. (or that would limit you to 6 breakers total).  The main breaker can be >= the feeder breaker size. 
Upgrade the feeder breaker to 60A, or if cable rating allows, 70A.  


Now you have 120-140A of power in that building, which is what you are after.  This will take a little bit of thought to provision power across 2 panels, but the savings are considerable over a trench-and-conduit job.