Joint communication and radar (JCR) waveforms with fully digital baseband generation and processing can now be realized at the millimeter-wave (mmWave) band. Prior work has proposed a mmWave wireless local area network (WLAN)-based JCR that exploits the WLAN preamble for radars. The performance of target velocity estimation, however, was limited. In this paper, we propose a virtual waveform design for an adaptive mmWave JCR. The proposed system transmits a few non-uniformly placed preambles to construct several receive virtual preambles for enhancing velocity estimation accuracy, at the cost of only a small reduction in the communication data rate. We evaluate JCR performance trade-offs using the Cramer-Rao Bound (CRB) metric for radar estimation and a novel distortion minimum mean square error (MMSE) metric for data communication. Additionally, we develop three different MMSE-based optimization problems for the adaptive JCR waveform design. Simulations show that an optimal virtual (non-uniform) waveform achieves a significant performance improvement as compared to a uniform waveform. For a radar CRB constrained optimization, the optimal radar range of operation and the optimal communication distortion MMSE (DMMSE) are improved. For a communication DMMSE constrained optimization with a high DMMSE constraint, the optimal radar CRB is enhanced. For a weighted MMSE average optimization, the advantage of the virtual waveform over the uniform waveform is increased with decreased communication weighting. Comparison of MMSE-based optimization with traditional virtual preamble count-based optimization indicated that the conventional solution converges to the MMSE-based one only for a small number of targets and a high signal-to-noise ratio.