I'm not super familiar with the jpeg format, but iirc h.264 uses 16x16 blocks, so if jpeg is the same then padding of 16px on all sides would presumably block all possible information leakage?

Except the size of the blocked section ofc. E.g If you know it's a person's name, from a fixed list of people, well "Huckleberry" and "Tom" are very different lengths.