The phase diagram of the three-dimensional Ising system with intrasurface J_s and surface-bulk J_^ interactions modified with respect to the bulk exchange interaction J_s is studied by the use of an effective field method within the framework of a single-site cluster theory. In this investigation, the new parameter J_^ strongly affects the phase diagram obtained recently in the case J_^=J_s. Indeed, our calculations reveal some qualitatively interesting features for the surface when the bulk is ordered or disordered. Surface magnetisations are also investigated for different values of the ratio R=J_s /J_B