IMHO there is no requirement to disconnect the old rod or to locate it and make sure that it is more than 6 feet from any other rods. If you can't find it just leave it connected and install whatever is required as if that rod were not there.
250(A)(3) is the relevant section, and it says 'If multiple...electrodes are installed to meet the requirements of this section, they shall not be less than 1.8m apart'. I read this to mean that an electrode which is too close simply doesn't count to meet the requirements, not that it needs to be removed.
Think about the physics. Ground rods that are close together are not as effective as ground rods that are spread apart, but if you had two ground rods installed side by side touching each other that would be no worse than a single ground rod.
-Jon