This paper presents a powerful analytical procedure that combines the higherorder boundary condition approximation and the modified Malyuzhinets technique to treat electromagnetic scattering from arbitrarily angled wedge-like composite configurations with penetrable faces. The configuration may be either a perfectly conducting wedge covered with a dielectric/ferrite material or a composite wedge with strongly absorbing faces, so that the field transmitted directly through the configuration can be neglected. With special emphasis on retaining the passivity of the media interfaces, boundary conditions on the wedge faces are approximated using impedance conditions with higher-order field derivatives. To close the formulation of the diffraction problem, a class of contact conditions is described that ensures the uniqueness and reciprocity. A general solution which is valid for passive boundary conditions of arbitrary odd orders is derived in an explicit form as the Sommerfeld integral involving arbitrary constants. By choosing these constants in such a way as to fit its analytical behavior for kr ≪ 1 to the exact one obtained by directly integrating the Maxwell equations, a specific form of the contact conditions can be deduced that reflects the design features of the configurations near their edges and forces the solution to be an asymptotic one for vanishing thicknesses of the coatings or skin layers. As a result, the unique and reciprocal closed-form solution associated with third-order boundary conditions is presented which uniformly describes the diffraction of an H- polarized plane electromagnetic wave by a variety of the composite wedge-like structures with penetrable faces, extending thereby Malyuzhinets' solution for an impedance wedge to more sophisticated boundary conditions.