I don't know any reason why a supplemental grounding electrode couldn't be bonded to a subpanel's grounding bus.
That wouldn't meet code..Why install a ground rod at the sub panel? All you have to do is run a #6 to a inter system bonding bridge from the ground bar of the sub.
The distance isnt an issue, the only issue is that your phone and catv are bonded to the electrical system grounding electrode system...Thanks to you all for taking the time to answer my question. With your answers and with some extra Internet research (that your answers triggered), I have decided to use the 6 AWG to bond a new ground rod to the old ground rod. I'll then ground my phone and cable to this new rod before having them enter the building. I found that the gap between the driveway and the garage slab should be wide enough to dig down a few inches to bury the wire (and I'll tunnel under the sidewalk). That way, I won't have the ground wire exposed to accidental cuts along the side of my house. With a few hundred dollars in materials and a lot of shoveling, I'll feel confident that everything is grounded properly. Plus, I'll have a bonus of a 2nd ground rod for my house. I'll bury the bonding wire 18 inches deep whenever possible to avoid future rototillers, lawn aerators, etc. (By the way, I initially said the distance between the old and new rod would be 30 or 40 feet. I was way off. When I measured it, the distance was 80 feet. I don't think this extra distance matters much in the effectiveness of the bond between the rods, but please let me know if I'm wrong.) Thank you all again.