In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order $p$ of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order $p+3$ (for $p\ge 3$) and have extended it to a generalized $(m+2)$-step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to $p+3m$, $m\in \mathbb{N}$. We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations.