Millimeter-wave (mmWave) multiple-input multiple-out (MIMO) systems relying on lens antenna arrays are capable of achieving a high antenna-gain at a considerably reduced number of radio frequency (RF) chains via beam selection. However, the traditional beam selection network suffers from significant performance loss in wideband systems due to the effect of beam squint. In this paper, we propose a phase shifter-aided beam selection network, which enables a single RF chain to support multiple focused-energy beams, for mitigating the beam squint in wideband mmWave MIMO systems. Based on this architecture, we additionally design an efficient transmit precoder (TPC) for maximizing the achievable sum-rate, which is composed of beam selection and beamspace precoding. Specifically, we decouple the design problems of beamspace precoding and beam selection by exploiting the fact that the beam selection matrix has a limited number of candidates. For the beamspace precoding design, we propose a successive interference cancellation (SIC)-based method, which decomposes the associated optimization problem into a series of subproblems and solves them successively. For the beam selection design, we propose an energy-max beam selection method for avoiding the high complexity of exhaustive search, and derive the number of required beams for striking an attractive trade-off between the hardware cost and system performance. Our simulation results show that the proposed beamspace precoding and beam selection methods achieve both a higher sum-rate and a higher energy efficiency than its conventional counterparts.