Sweet potatoes are typically grown from slips (cuttings) and not 'seeds' or pieces of tuber. So, you did it like it's usually done. I am not familiar with timing gardens in the south, so I don't know what your timetable is for good tuber production. But, if you got it in at the right time, you should have success.

Oh, doing the usual/right things is unusual with my practices, LOL

So... one plant only yields ONE tuber, right?

If I harvest the tuber, can I re-root the plant and make it continuously grow?

chudh - For future endeavors, you can take one potato, and submerge about the bottom third of it in a glass of water. You can stick toothpicks in the potato about a third of the way down and rest the toothpicks on the glass.
The potato will begin to sprout. After the sprouts are about 6 inches long, you can pop them off of the potato and put them in water and they will make roots. These small plants are called slips, and you can make multiple slips from one potato. In fact, I have a barrel of sweetpotato plants growing in my garden right now that all came from one potato. I didn't throw it out when I finished and now my kitchen window is filled with potato vines. You will have to add more water to your glass from time to time. Easy to grow.

Sweet potatoes do not form tubers. The storage unit is a modified root. so if you planted it and it has roots you have sweet potatoes. Once the vine sizes up they began storing food in the roots. Depending on time and nutrients, one to a couple of dozen roots will swell enough to be useful as a potato. One possible downside, ornamentals have roots but often they are tough and stringy, and not really food quality. But you can save them and regenerate plants next year just like you would with a food quality potato.