In order to mitigate the significant path loss encountered in millimeter wave systems, large-scale antenna arrays are employed. However, in such systems, the primary hurdles revolve around maintaining low-cost hardware and minimizing power consumption. Moreover, the hardware is entrusted with the complex task of computing high-dimensional optimal matrices for hybrid precoding design utilizing the Orthogonal Matching Pursuit (OMP) algorithm. In this paper, we suggest a new modified version of OMP as efficient algorithm for hybrid precoding that effectively improves system performance by enhancing spectral efficiency. In fact, we use the spatial arrangement of mm-wave channels to frame the precoding challenge as a sparse reconstruction problem. Employing the concept of basis pursuit, we introduce the modified OMP technique, which aims to approximate the optimal fully digital precoder. Our numerical results demonstrate the superiority of the new modified OMP approach with respect to some existing precoding methods.