The general method that is often used here in the U.K. is to pass a current through the ground and then measure the potential using an auxiliary electrode.
Call the existing grounding system X. Drive a ground rod, Y, into the earth at some distance from any of the existing ground rods. The reistance areas must not overlap.
Connect an adjustable low-voltage source (we usually use about 25 to 40V AC) between X and Y with an ammeter in series to monitor the current.
Now drive an auxiliary electrode Z at a point roughly mid-way between X and Y, and connect a voltmeter between X and Z. The ground system reistance is then simply E/I by Ohm's Law.
To check that the rod resistance areas are not overlapping, move rod Z about 10% of the X-Y distance closer to X and repeat the test. Do the same with Z moved about 10% closer to Y. If all three readings are approx. the same, then take this as the system earth resistance. If the readings differ substantially, then you will need to move ground rod Y farther away and start over.
To obtain accurate readings, the original ground system X must be disconnected, otherwise any other interconnected paths to ground will affect the reading.