Aiming for seamless satellite communications on the move (SOTM) between vehicles and new low-earth orbit satellite constellations, this paper presents a low-power beam-steering antenna system with a novel beam handover algorithm. The proposed antenna uses a 3D-printed Luneburg lens designed with a planar focal surface. The lens is illuminated with a hexagonal array of circularly polarized elliptical patch antennas operating at 11.7 to 12.7 GHz and fed with a novel cascaded beamformer network to achieve a gain of 21.5 (± 0.5) dBi with a total efficiency > 75% and side lobe levels below 15 dB. A highly efficient and novel handover algorithm has been developed to achieve a fast beam-steer of ±55⁰ in both azimuth and elevation planes. The proposed handover procedure has been demonstrated within an anechoic chamber. This paper has applications in ensuring a smooth beam-steer while maintaining uninterrupted connectivity between fast-moving satellites and vehicles on the road using the proposed antenna.