The use of capacitive metal meshes consisting of square patches is well established in the design of millimeter-wave optical devices such as filters, half-wave plates, and graded-index lenses. The main property of these meshes, the capacitance, is controlled by the size of the square patch in the mesh’s unit cell up to a limit imposed by the minimum feature size of the manufacturing process. We report the results of a study into the feasibility of increasing the capacitance beyond this limit by extending fingers into the adjacent unit cell. This is shown to increase the range of admittance characteristics available with the meshes and how this might be useful for the implementation of low-pass filters and artificial dielectrics is demonstrated.