<p>Progressive-iterative approximation is an efficient tool for data interpolation and approximation in CAD systems. For greater flexibility, a local progressive-iterative approximation method was proposed in [Lin HW, Local progressive-iterative approximation format for blending curves and patches, Computer Aided Geometric Design 27 (2010) 322-339] by adjusting only a subset of the control points. To speed up the convergence rate, this paper presents a family of hybrid local progressive-iterative approximation methods based on a high-order convergent iterative method for computing the inverse matrix. We demonstrate that the proposed accelerated local iterative procedure is convergent and the limit curve interpolates the corresponding adjusted data points. Furthermore, we generalize the proposed iterative format of blending curves to the surface case. The iterative format is represented in matrix form to avoid the Kronecker product computation, which significantly increases computational cost and could potentially produce an ill-conditioned matrix. Several numerical examples, including adaptive fitting, are given to validate that our method outperforms the existing approaches, particularly with larger datasets.</p>