In this article, we address the problem of reducing the number of required samples for Spherical Near-Field Antenna Measurements (SNF) by using Compressed Sensing (CS). A condition to ensure the numerical performance of sparse recovery algorithms is the design of a sensing matrix with low mutual coherence. Without fixing any part of the sampling pattern, we propose sampling points that minimize the mutual coherence of the respective sensing matrix by using augmented Lagrangian method. Numerical experiments show that the proposed sampling scheme yields a higher recovery success in terms of phase transition diagram when compared to other known sampling patterns, such as the spiral and Hammersley sampling schemes. Furthermore, we also demonstrate that the application of CS with an optimized sensing matrix requires fewer samples than classical approaches to reconstruct the Spherical Mode Coefficients (SMCs) and far-field pattern.