(n.b. the only latitudinal Great Circle is the equator).
The infinite number of solutions I had in mind were, to take your original problem, find a line of latitude close to the South Pole that is exactly 1/2 a mile in length, then start 1 mile north of it so your journey takes you around the pole twice. Then find a line of latitude that is exactly 1/3 a mile in length, one that's 1/4 mile, etc etc. It all gets to be a bit of a blur when you're starting off just a little bit more than 1 mile from the pole.
The book was a very old one, and it's been so long that I couldn't tell you the title off the top of my head. I'll get a chance to look for it at my parents' place in a month or two, but in the meantime Martin Gardner books are worth mining.