[Taf]

My late Uncle did this job professionally. He used a ground anchor, 2 Steel A-frames and a hand winch to lift the concrete out of the ground. No digging, just soak the earth and lift. If the post stump was still sound, he used another spiral ground anchor to attach the cable to it. Otherwise, he had an expanding anchor that gripped the hole the post had been in.

Fence posts tend to rock back and forth, so if you have to resort to digging it out, dig only from one SIDE, never front or back. This will leave compacted earth that will make a new concrete base stable.

He taught me a great trick for fitting new posts. Wrap the bottom of the post in 2 plastic bags, each held in place with tape. Place this bagged end into position in the hole, but stood on a couple of inches of gravel. Concrete into place, and once cured, you lift the post out, remove the bags and drop the post back into the hole. When the post needs to be replaced, you just lift it out and drop in another (posts may be "standard" sizes, but sometimes you might need to trim the new one down a touch). The gravel underneath allows water to drain away, extending the life of the posts.