In this paper we investigate the precoder design problem of a preprocessing-aided spatial modulation multiple-input-multiple-output (PSM-MIMO) system with quadrature-phase-shift-keying (QPSK) signaling that facilitates hardware implementation using one-bit digital-to-analog-converters (DACs) and quantized phase shifters. We formulate the new design problem and propose an efficient algorithm based on a combination of alternating minimization method and mathematical programming with equilibrium constraints (MPECs). The bit-error-rate (BER) performance of the proposed one-bit precoder in comparison to a number of benchmark PSM-MIMO precoders is quantified via numerical simulations.