The box you're changing out to is a temporary power outlet panel. To bring it up to code you need a separate grounding conductor going back to the main panel. Do not add another ground rod and call it good unless you want to set yourself up for a major hurt in the future.
Since this is probably replacing a very old installation, I believe it is considered grandfathered and you can tie your ground wires inside the new panel into the neutral buss bar inside the new power panel. I would check with your local electrical inspector before doing this however.
The safest thing to do would be to change out the 3 wire feed with a 4 wire feed.