Yeah, 3 to 6 inches seems pretty short. It might work alright with some rooting formula. I personally wouldn't go any shorter than maybe about 8 or 9 inches though.

Really short ones did seem to fail a bit more in the past for me, but I'd probably have to do an experiment to know for sure what the optimum length is.

About a foot just seems about right, since it gives you space to cut, remove a few leaves, and still have a few at the end, then stick it in the ground.

Sweet potatoes are pretty easy to grow though. You can actually just work the vines back into the soil with a hoe and quite a few will pop back up. They'll also pop up out of your compost after harvest for a few weeks or months.