Since the classical weighted essentially non-oscillatory (WENO) scheme is proposed, various improved versions have been developed, and a typical one is the WENO-Z scheme. Although better resolution is achieved, it is shown in this article that, the result of WENO-Z scheme suffers evident distortion in the long-time simulation of the linear advection equation. In order to fix the problem of WENO-Z scheme, a symmetry-preserving mapping method is proposed in this article. In the original mapping method, the weight of each substencil is used to map, which is demonstrated to cause asymmetric improvement about a discontinuity. This asymmetric improvement will lead to a distorted solution, more severe with longer output time. In the symmetry-preserving mapping method, a new variable related to the smoothness indicator is selected to map, which has the same ideal value for each substencil. Using the new mapping method can not only fix the distortion problem of WENO-Z scheme, but also improve the resolution. Several benchmark problems are conducted to show the improved performance of the resultant scheme.