We present a path planning algorithm for a polyhedral robot with six degrees of freedom (6DOF) and a static obstacle. The planner consists of a dual-tree RRT algorithm that uses a novel local planner. A local planner tests if two robot configurations can be connected by a simple path. Ours searches a 3DOF subspace of the robot/obstacle configuration space, whereas prior planners search the line segment that connects the two configurations. Empirical evidence suggests that the benefit of the 3DOF search outweighs the cost. Our planner outperforms prior planners on problems with narrow channels and performs comparably (shorter paths, similar running times) on other problems