Spatial channel covariance information can replace full knowledge of the entire channel matrix for designing analog precoders in hybrid multiple-input-multiple-output (MIMO) architecture. Spatial channel covariance estimation, however, is challenging for the hybrid MIMO architecture because the estimator operating at baseband can only obtain a lower dimensional pre-combined signal through fewer radio frequency (RF) chains than antennas. In this paper, we propose two approaches for covariance estimation based on compressive sensing techniques. One is to apply a time-varying sensing matrix, and the other is to exploit the prior knowledge that the covariance matrix is Hermitian. We present the rationale of the two ideas and validate the superiority of the proposed methods by theoretical analysis and numerical simulations. We conclude the paper by extending the proposed algorithms from narrowband massive MIMO systems with a single receive antenna to wideband systems with multiple receive antennas.