The 3D characterization of radiating structures is a frequently encountered, and yet challenging, procedure. The theoretical sampling criterion often leads to prohibitive measurement duration, especially for antennas embedded on electrically large structures. However, the position of the antenna under test strongly influences the sampling requirements. By appropriately positioning the antenna with respect to the measurement system, it is possible to promote the low spatial frequency part of its spherical harmonic spectrum. We present a post-processing optimization algorithm enabling a systematic and accurate determination of the antenna best position from a given measurement dataset with no prior knowledge. This sooptimized position is exploited to improve the characterization of the antenna radiation pattern without resorting to any additional measurement sample. The proposed algorithm is validated for both near-field simulations and far-field measurements on various radiating structures, including an antenna embedded on a satellite model.