In this paper we introduce a new Jacobian-free Steffensen-type iterative process aimed at enhancing the convergence order of existing methods. Starting from a method of order \(p\) , we append a third computational stage to obtain a scheme of order \(p+3\) , while still avoiding explicit Jacobian matrix evaluations, which are usually the most expensive part of the iteration. A parametric multipass family of Jacobian-free Steffensen-type methods is derived, and it is shown that appropriate choices of the free parameters allow us to systematically raise the convergence order. The construction makes use of auxiliary functions and weighting operators that modify the error equation and improve the convergence behaviour while keeping a fully Jacobian-free structure and a controlled computational cost. The new procedures are tested on several types of problems, such as nonlinear systems, integral equations and nonlinear partial differential equations (PDEs), in order to assess their range of applicability. Numerical experiments, together with comparisons against existing schemes of similar convergence order, demonstrate the competitiveness and computational efficiency of the proposed family.
