marlinarchy - I was a bridge contractor many years ago. IHMO, the first technique is to try jetting around the perimeter of a pile. Success will depend on the depth of pile embedment and the soil, jetting works well with sandy soil or "mud". The pump used for this does not have to be very sophisticated. A 2" centrifugal pump with the jet being a length of Schedule 40, steel pipe is an inexpensive way to start. Nominal 1 1/4" pipe is a good match to use as the jet for a 2" pump, a good compromise of water velocity and flow volume. This outfit is small enough to be worked by hand. You probably will need a crane, or winch to provide a small upward force on the piling while the jetting is going on. Can takes a lot of patience.
If the piles are larger, or deeper, try a 3" or 4" pump. Another trick is to modify the jetting pipe so that compressed air is injected into the water steam at the top of the pipe. The combination of water / air may improve results. Try just water first.
With care, jetting is not going to damage anything, even if it does not work on your project. It is a lot less damaging than using a pile extractor, or just jacking on the piling alone.
If jetting is not practical, or does not work, the next technique I would use would be a small earth auger parallel to and a short distance from a pile.
If you have any questions on this, I try to answer them.
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