We present a local convergence analysis of a Modified Halley-Like Method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet-derivative [Wang, X., Kou, J., Convergence for modified Halley-like methods with less computation of inversion. J. Diff. Eq. and Appl. 19(9) (2013), 1483-1500.]. Numerical examples are also provided in this study.