You can do slightly better than that. Go in a straight line for sqrt(2) miles in, let's say, the northwest direction, then travel south to the west edge of the 1 mile radius circle surrounding your starting point, walk along the southern half of the circle, and then go a mile north, so that you finish sqrt(2) miles northeast of where you started.
I haven't been able to prove this is optimal yet, or tried to work out the expected distance version, so those are still things someone could try.
