It really depends what sort of soil grower you want to be. If the container is sufficiently large (15+ gal), you can plant your new plant in the same pot and leave the first stump there to decompose. If you have a healthy soil web (fungus, microbes, worms, decomposer insects, etc), the roots will decompose over a few weeks/months and will eventually pull out with little effort.
For smaller pots, I would try to amend and/or compost the used soil like sct2020 mentioned.