Millimeter-wave (mmWave) joint communication-radar (JCR) simultaneously realizes a high data rate communication and a high-resolution radar sensing for applications such as autonomous driving. Prior JCR systems that are based on the state-of-the-art mmWave communications hardware, however, suffer from a limited angular field-of-view (FoV) and low detection rate for radars due to the employed directional beam. To address this limitation, we propose an adaptive and fast combined waveform-beamforming design for mmWave JCR with a phased-array architecture. We present a JCR beamformer design algorithm that permits a trade-off between communication data rate and radar successful recovery rate in the angular domain. We show that distinct radar measurements can be obtained with circulant shifts of the designed JCR beamformer for compressed radar sensing. Numerical results demonstrate that our JCR design enables the angle-of-arrival/departure estimation of short-range radar targets with a high successful recovery rate and a wide FoV at the expense of a slight loss in the communication rate.