In this paper, we tackle a more general coordinate-free investigation of the second approximation Matsumoto metric. Specifically, we establish various geometric objects corresponding the second approximation Matsumoto metric L̃(x,y) = L + B + B2/L + B3/L2  in terms of the objects of L with a Finsler structure (M,L) and a one form B. Here we consider L is Finslerian and so we call L̃ the generalized second approximation Matsumoto metric. We describe the metric tensor of L̃ and its non-degeneracy. We limit the one form B to a one form associated with a concurent π-vector field in order to determine the geodesic spray, Barthel, and Berwald connections of L̃(x,y). Next, we compute the Barthel curvature of L̃.