One last proposition. Cushman's answer was based on dividing the circle into four equal arcs, or drawing a square around the circle:


My last answer was based on dividing the circle into 3 parts, and the path was longer than Cushman's. Dividing it into 2 parts wouldn't be possible, he would just keep walking out to infinity never reaching the point where he needs to make his turn. so...
2= infinity
3= 7.55
4= 6.55

so what would happen if you drew a pentagon around the circle.

he would walk out from the tree for 1 km plus a short distance smaller than .414 km. then he'd turn, meet the circle and walk four fifths of it, then walk another short distance off on a tangent.

I'm too tired to do any more math, but I doubt it will be shorter than 6.55. it will probably just get higher and higher for inifinity the more you divide up the circle but never reaching the circumference of the circle added to the radius. In fact, now I'm positive it will do that without doing the math. I think cush's is the shortest possible. Good job, cushman.