For control systems whose uncontrolled solutions are periodic (or more generally recurrent), there are geometric tools, developed in the 1980s, that assess controllability based on a Lie algebra rank condition, under the assumption that the control set contains zero in its interior. Motivated by solar sails control, the present study explores the case where zero is rather on the boundary of the control set. More precisely, it investigates the controllability of fast-oscillating dynamical systems subject to positivity constraints on the control variable, i.e., the control set is contained in a cone with vertex at the origin. A novel sufficient controllability condition is stated, and a constructive methodology is offered to check this condition, and to generate the controls, with values in the convex cone, that move, at first order, the slow state to an arbitrary direction of the tangent space. Controllability of a solar sail in orbit about a planet is analysed to illustrate the developments. It is shown that, given an initial orbit, a minimum cone angle parametrising the control set exists which satisfies the sufficient condition.