I'll try to be comprehensive...
The CEE is sufficient to serve as the premises grounding electrode system by itself*, but currently it does not qualify for that because it is connected to a subpanel. Because of where it's currently connected, it currently qualifies only as an auxilliary electrode (250.54). It is not a code violation to have it connected there, but it's not advisable, and ...
The ground rod is insufficient to serve as the grounding electrode system by itself (250.53(A)(2)), so the grounding electrode system is not up to code.
Relocating/extending the GEC for the CEE to terminate anywhere between the outside service disconnect and the service point turns it into a premises electrode and fixes the code violation of insufficient grounding. Doing so makes the ground rod superfluous, but since it is there it must remain bonded to the rest of the system. (250.50) As far as the NEC is concerned, the ground rod GEC termination can stay where it is, or be moved to the service disconnect, or be bonded to the CEE GEC. From what I've gathered on this forum, the utility may either insist that it be moved from the meter socket or require that it stay there, depending on who the utility is and what part of the country you're in. Yup, utilities are that arbitrary.
*If there is an underground metal water pipe on the premises, it's also required to be bonded to the service neutral or one of the other electrodes.