Geometric tools for the assessment of local controllability often require that the control set has the origin within its interior. This study gets rid of this assumption, and investigates the controllability of fast-oscillating dynamical systems subject to positivity constraints on the control variable, i.e., the convex exterior approximation of the control set is a cone with vertex at the origin. A constructive methodology is offered to determine whether the averaged state of the system with controls in the exterior convex cone can be locally moved to an arbitrary direction of the tangent manifold. The controllability of a solar sail in orbit about a planet is analysed to illustrate the contribution. It is shown that, given an initial orbit, a minimum cone angle exists which allows the sail to move slow orbital elements to any arbitrary direction. This is an extended version of a document by the same authors untitled “ Effective Controllability Test for Fast Oscillating Control Systems. Application to Solar Sailing”, preprint hal-03185532v1.