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In this paper, we introduce a superconvergent approximation method that employs radial basis functions (RBFs) in the numerical solution of conservation laws. The use of RBFs for interpolation and approximation is a well developed area of research. Of particular interest in this work is the development of high order finite volume (FV) weighted essentially non-oscillatory (WENO) methods, which utilize RBF approximations to obtain required data at cell interfaces. Superconvergence is addressed through an analysis of the truncation error, resulting in expressions for the shape parameters that lead to improvements in the accuracy of the approximations. This study seeks to address the practical elements of the approach, including the evaluations of shape parameters as well as hybrid implementation. To highlight the effectiveness of the non-polynomial basis, in shock-capturing, the proposed methods are applied to one-dimensional hyperbolic and weakly hyperbolic systems of conservation laws and compared with several well-known FV WENO schemes in the literature. In the case of the non-smooth, weakly hyperbolic test problem, notable improvements are observed in predicting the location and height of the finite time blowup. The convergence results demonstrate that the proposed schemes attain notable improvements in accuracy, as indicated by the analysis of the reconstructions. We also include a discussion regarding extensions to higher dimensional problems, along with convergence results for a nonlinear scalar problem.