The only way a GEC can be a #6 is if the only electrodes are rods. 250.66(A) says " that portion of the conductor that is the sole connection to the grounding electrode shall not be required to be larger than 6 AWG".
If this is going on to another electrode, the GEC must be sized to the best electrode you have in the system, up to what you get from the 250.66 table.

From the handbook

[Linked Image from gfretwell.com]

Greg Fretwell