[dpifke]

Multiple grounding rods (electrodes) are permitted by § 250.52, but they have to be bonded together and kept away from other (non-bonded) grounding systems:

Electrode Spacing. Where more than one of the electrodes of the type specified in 250.52(A)(5) or (A)(7) are used, each electrode of one grounding system (including that used for strike termination devices) shall not be less than 1.83 m (6 ft) from any other electrode of another grounding system. Two or more grounding electrodes that are bonded together shall be considered a single grounding electrode system.

http://thenecwiki.com/2021/02/article-250/

That said, it would be very strange to add another grounding rod just because you added another breaker or outlet—I can't think of any circumstance where that would be required, unless the existing service entrance was not properly grounded, in which case you have bigger problems.

[amluto]

That section AIUI is about “grounding electrodes” that are intended for use as part of the required main ground. NEC 250.54 says you may install an auxiliary grounding electrode if you are so inclined without following all those rules. Other sections of the NEC explicitly allow burying things like rigid metallic conduit underground, which is a lot like a grounding rod attached to a branch circuit or feeder.