An algorithm called MUSIC-like algorithm was originally proposed as an alternative method to the MUltiple SIgnal Classification (MUSIC) algorithm for direction-of-arrival (DOA) estimation. Without requiring explicit model order estimation, it was shown to have robust performance particularly in low signal-to-noise ratio (SNR) scenarios. In this letter, the working principle of a relaxation parameter β, a parameter which was introduced into the formulation of the MUSIC-like algorithm, is provided based on geometrical interpretation. To illustrate its robustness, the algorithm will be examined under symmetric α-stable distributed noise environment. An adaptive framework is then developed and proposed in this letter to further optimize the algorithm. The proposed adaptive framework is compared with the original MUSIC-like, MUSIC, FLOM-MUSIC, and SSCM-MUSIC algorithms. A notable improvement in terms of targets resolvability of the proposed method is observed under different impulse noise scenarios as well as different SNR levels.