[dfan]

This is a good advertisement for the approach I came up with eventually: if you don't see a trivial way to the next square, find a sequence that will lead to it that starts in the lower right quadrant, then go to that quadrant and get to that square. (The lower right quadrant is the center ring in this diagram.)

I'm used to a version with pawns that involves less backtracking, so it took me a little while to stop trying to be "smarter" and just be okay with continually returning to home base.