For a water- or slurry-filled hole, there is a device called a SoniCaliper that is lowered on a cable from a pulley at the top of the hole and measures the radius in all directions every 2 feet (or other increment) of depth. Because the device hangs plumb, the deviation of the hole center from a plumb line can readily be determined.
For a low-tech approach, you can build an inverted plumb bob. This looks like a cylindrical bird cage a little smaller than the hole, with skids or wheels on the sides. A string is attached to the center of the cage, and the cage is lowered down the hole by a pair of ropes. You can stop at any desired depth, attach a cylindrical buoy to the string, and adjust the string so the buoy is almost fully submerged. The center of the buoy is directly above the center of the hole at that depth. You can measure its distance from the sides of the hole in the N-S and E-W directions.
If the hole is dry, you can use the inverted plumb bob by placing the equivalent of a large hollow-centered cake pan over the hole. The bob wire goes up through the center hole in the pan and attaches to a frame straddling the "cake pan" and supported on floats, floating on water in the pan.
