Non-Gaussian Harmonizable Fractional Stable Motion (HFSM) is a natural and important extension of the well-known Fractional Brownian Motion to the framework of heavy-tailed stable distributions. It was introduced several decades ago; however its properties are far from being completely understood. In our present paper we determine the optimal power of the logarithmic factor in a uniform modulus of continuity for HFSM, which solves an open old problem. The keystone of our strategy consists in Abel transforms of the LePage series expansions of the random coefficients of the wavelets series representation of HFSM. Our methodology can be extended to more general harmonizable stable processes and fields.