In this paper, we study a variation of second order periodic averaging that allows an asymptotic reconstruction of the " fast " variable. Our goal is to present a simpler method to obtain a second order estimate based on changing the initial condition of the averaged system and approaching a " moving " average of the solutions instead of the solutions themselves. This idea was already present in a publication by the first author, in the context of solving an optimal control problem with averaging techniques and application to low thrust orbit transfer. Namely, adding well suited second order terms to the average system, properly choosing the initial condition for the average system and measuring errors " in the mean " instead of point-wise yields an errors of order 2 with respect to ε on the slow variable and of order 1 for the fast variable.