The atomic nucleus can be used as a laboratory for testing the Standard Model of the electroweak interaction at the precision frontier. Some particular types of nuclear $\beta$ decays provide a promising tool to probe the possibility of additional quark generations as well as of exotic weak currents beyond the standard $V-A$ theory. Over the past 40 years, the experimental quantities that are required for performing these tests have been measured to high-precision with rare-isotope beam facilities around the world. Currently, the limiting factor in these studies is rather the corrections due to the imperfection of the isospin symmetry and the radiative effects which depend on details of nuclear structure. Our research focuses on the isospin-symmetry breaking correction within the shell-model framework for superallowed Fermi and Gamow-Teller $\beta$ decays of low‐to‐medium‐mass nuclei. As an improvement, we extend the formalism to include the higher order contribution which has always been neglected. We also develop a new method for fitting the depth and the radius of the Woods-Saxon potential to ensure the correctness of asymptotic radial wave functions. The results are compared to previous calculations and their implication for the Standard Model is discussed.