Millimeter-wave (mmWave) and multiple-input multiple-output (MIMO) combination technologies have attracted extensive attention from both academia and industry for meeting future communication challenges and requirements. As a viable option to deal with the trade-off between hardware complexity and system performance, hybrid analog/digital architectures are regarded as efficient mmWave MIMO transceivers. While acquiring channel state information (CSI) is a challenging task to design the optimal beamformers/combiners, especially in mmWave communications due to a lot of challenges. Fortunately, the sparse nature of the channel allows to leverage the compressed sensing (CS) tools and theories. However, the critical challenge to develop a CS-based formulation for estimating the mmWave channel is the codebook design (sensing matrices) and its pilot symbol numbers. In this paper, we proposed a multistage CS-based algorithm to estimate the channel explicitly using pilot and data symbols which enable increasing the number of measurements to enhance the estimation accuracy and maximize the spatial diversity by reducing the overlapping between training beams. Simulations confirmed that our proposed method has the best results compared to the existing methods based on codebook schemes.