The present paper investigates the approximation of special monogenic functions (SMFs) in infinite series of hypercomplex Hasse derivative bases (HHDBs) in Fréchet modules (F-modules). The obtained results ensure the existence of such representation in closed hyperballs, open hyperballs, closed regions surrounding closed hyperballs, at the origin, and for all entire SMFs (ESMFs). Furthermore, we discuss the mode of increase (order and type) and the Tρ-property. This study enlightens several implications for some associated HHDBs, such as hypercomplex Bernoulli polynomials, hypercomplex Euler polynomials, and hypercomplex Bessel polynomials. Based on considering a more general class of bases in F-modules, our results enhance and generalize several known results concerning approximating functions in terms of bases in the complex and Clifford settings.