I've used post hole diggers and a sledge hammer around loose/partially broken posts to minimize digging, ease extraction, and pack the hole adequately. Dig a deep hole shaped like the letter C tightly around the post, and it will be much easier to remove the old post.
To pack the new post in this narrow hole, use the handles of the post hole diggers (it is awkward, but works very well). It is important to pack small batches of dirt in several layers. Near the top of the hole, use a sledge hammer, but don't swing it like a hammer. Use it like an actual tamper in an up and down motion, striking the surface with the top of the head. This will set the fence or mailbox post so well that it will usually feel like concrete was used. This approach will not work well in poorly drained areas that tend to be saturated with water.
Years ago, I heard a fence installer explain that concrete would speed the decomposition of the wood post due to the lime content. Can any of the pros verify that claim?