The problem of the wave fields excited by a gap source at the edge of an inhomogeneous, magnetized plasma with a pair of lower-hybrid resonance layers present and bounded by conducting walls is solved. The approach used is that of a solution as a sum of multiply-reflected extraordinary mode and ion-thermal resonance cones as an alternative to the guided-wave mode approach. This is achieved by dividing up the plasma into cross-sections where WKB solutions are valid, and into lower-hybrid resonance layers where asymptotic methods are employed and connexion coefficients between each region are obtained. A diagrammatic scheme for writing the solution is introduced which can in principle be used to write down the solution for any problem of this general type once the connexion coefficients across a resonance layer have been determined via asymptotic analysis. This allows a determination in great detail of the structure and properties of the resonance cones in our model and the way they transform across the back-to-back hybrid layers. Evanescent resonance cones are found to exist in the high-density region between the hybrid resonance layers and tunnel through to the other side, maintaining their general structure if the layer is relatively thin.