This paper introduces a comprehensive investigation on the possibility to tailor the ring focus antenna's beamwidth by manipulating the shape of the sub-reflector, and more specifically its eccentricity, determining the location of the focal points of the dual reflector system. Eccentricity variation affects the sub-reflector's reflected field and thus the main reflector's illumination. The examined antenna configuration includes a main reflector with a diameter of 13.5 m, a sub-reflector with a diameter of 1.35 m, and employs a linearly polarized Gaussian beam as feed. Changing the sub-reflector's eccentricity from the original value of 0.6 to 0.66 reduces the directivity from 70 dB to 55 dB, widening the beamwidth from 0.05° to 0.45° at 26 GHz. This feature is of great interest for applications needing adjustable beamwidth, such as satellite location acquisition. The beamwidth switching may be implemented using a reconfigurable transmitarray providing two states equivalent to the two sub-reflector shapes.