He takes the cane with a small "Thanks." as he straps it back to his wrist - and then almost as soon as he hears Bucky walking away in one direction for his own investigation, he walks in the other, one hand outstretched to maintain his presence against the boundary, the other using his cane to make sure the path is clear for him.

Look, he just wants to see how far this strange limitation goes. And if he has to hop a fence by the side of the road to do it - well, he's not quiet, about the way he grunts and lifts himself with surprising speed over the much more real and interactable barrier, and keeps following it.