In this work we demonstrate the extended and generalized methodology for the design of Quad-Furcated Profiled Horns (Q-FPHs). Based on a design case of a 4 lambda(0) x 4 lambda(0) Q-FPH, we extract the Generalized Scattering Matrix (GSM) of the enlarged quad-furcated discontinuity and provide analytical expressions for its multimode feeding. Next, the four feeding and the upper common waveguide sections are optimized accordingly through Mode-Matching (MM). The high aperture efficiency levels delivered by the methodology are verified by full-wave simulations of the optimized design case and compared to the state-of-the-art which is thereby redefined.