Adding second ground rod. Is this an acceptable way of bonding?
NEC requires the bonding electrodes be connected via an approved method if joined together outside an approved assembly like the main disconnects bonding lugs.  In other words, you might have done it correctly assuming the main disconnect is actually in doors and not outdoors, and it had a free lug opened. If there were not any free lugs available then it would need to be secured via an approved method.

Adding two new ground rods ( 6 ft a part is Code ) would be ideal and leave the old one buried and unused.  Though as mentioned by Ecnerwal it would not hurt anything to leave it connected.  I say disconnect it as the next home owner will just be confused.

Edit

Additional info has came to my attention in that your drawing shows a box below the meter.  This is typical of a main disconnect.  Just to be clear, the panel inside would not be considered the main disconnect if there was already one outside.  If this is the case then connecting the grounding electrode would only be acceptable at the main disconnect and not the panel inside.

Also, Code only requires one 8 ft ground rod if the minumin 25 ohms is met.  It is easier to drive two ( Code says 6 ft a part ) and not bother testing if the 25 ohms is reached.