In this paper, we develop hybrid analog/digital precoding based on the fully-connected architecture for massive multiple-input multiple-output (MIMO) low earth orbit (LEO) satellite communications (SATCOM), by exploiting the statistical channel state information (CSI) at the transmitter. The hybrid precoder design is formulated as an energy efficiency (EE) maximization problem by considering both continuous and discrete phase shift networks for implementing the analog precoder. The resulting optimization problem is nonconvex and difficult to solve. To that end, first, we apply a closed-form tight upper bound to approximate the ergodic rate. Then, we adopt Dinkelbach’s algorithm and the iteratively weighted minimum mean-square error (WMMSE) method to obtain the fully digital precoders. After that, the alternating minimization and inexact majorization-minimization (MM) algorithms are utilized to compute the hybrid precoders. Simulation results show that the proposed algorithmic solutions achieve significant performance gains when compared to existing literature ones, especially in the case where the discrete phase shift network is employed for analog precoding.