For sweet potato plants to transplant to the garden, wait until several stems are at least six inches long and have multiple leaves. Remove these stems close to the old potato root, and remove all but the top two leaves. Set these "slips" in a glass of water, where they will root within two weeks. The rooted slips may be transplanted directly to the garden if the soil has warmed up, or you can put them in pots to hold for a few weeks until favorable planting conditions arrive.
Don't discard the old sweet potato, however. It will continue to produce stems and leaves, feeding off the carbohydrates stored within it last season. You can continue to take slips until you have as many as you like. Thereafter, you can harvest the leaves from the potato any time. They make delicious additions to salads, soups and stir frys. Use them as you would spinach. You can eat the stems, too, if they are not too tough. The original root will continue to produce leaves until all of its stored food is exhausted. At that point, it is ready for the compost pile. Until then, you should get several servings of sweet potato leaves.