For a control system with one fast periodic variable, with a small parameter measuring the ratio between time derivatives of fast and slow variables, we consider the Hamiltonian equation resulting from applying Pontryagin maximum principle for the minimum time problem with fixed initial and final slow variables and free fast variable. One may perform averaging at least under normalization of the adjoint vectors and define a "limit" average system. The paper is devoted to the convergence properties of this problem as the small parameter tends to 0. We show that using the right transformations between boundary conditions of the "real" and average systems leads to a reconstruction of the fast variable on interval of times of order 1/ε where ε is the small parameter. This is only evidenced numerically in this paper. Relying on this, we propose a procedure to efficiently reconstruct the solution of the two point boundary problem for nonzero ε using only the solution of the average optimal control problem.