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In this work, the reduced scalar potential is utilized to derive a finite element formulation capable of handling hysteretic nonlinear material laws. The Fréchet derivative is employed to deduce a quasi-Newton scheme in the weak form for solving the nonlinear partial differential equation that describes the magnetostatic field. Methods for evaluating the Jacobian are presented, and their performance is compared on a 3-D domain under uniaxial and rotational excitation. The numerical results demonstrate the necessity of a flexible approximation to overcome the non-uniqueness of the Jacobian at reversal points, which naturally occurs in hysteresis loops. Consequently, an exact or excessively localized evaluation would give rise to difficulties in states of material magnetization characterized by these reversal points.