Multiple signal classification (MUSIC) has been widely applied in multiple-input multiple-output (MIMO) receivers for direction-of-arrival (DOA) estimation. To reduce the cost of radio frequency (RF) chains operating at millimeter-wave bands, hybrid analog-digital structure has been adopted in massive MIMO transceivers. In this situation, the received signals at the antennas are unavailable to the digital receiver, and as a consequence, the spatial covariance matrix (SCM), which is essential in MUSIC algorithm, cannot be obtained using traditional sample average approach. Based on our previous work, we propose a novel algorithm for SCM reconstruction in hybrid massive MIMO systems with multiple RF chains. By switching the analog beamformers to a group of predetermined DOAs, SCM can be reconstructed through the solutions of a set of linear equations. In addition, based on insightful analysis on that linear equations, a low-complexity algorithm, as well as a careful selection of the predetermined DOAs, will be also presented in this paper. Simulation results show that the proposed algorithms can reconstruct the SCM accurately so that MUSIC algorithm can be well used for DOA estimation in hybrid massive MIMO systems with multiple RF chains.