Abstract <p>This paper presents two enhanced high-order weighted compact nonlinear schemes (WCNSs) for solving the compressible Euler equations. Based on the framework of the WENO–Z nonlinear interpolation technique, a novel global smoothness indicator of a three-point global stencil is designed to deal with the issue of order reduction at critical points. The WCNS scheme with the enhanced global smoothness indicator is called the WCNS–ZM scheme that ensures third-order convergence accuracy even in the presence of second-order critical points. To enhance the local accuracy, perturbation terms with a free parameter are introduced to the linear interpolation based on a three-point smooth stencil. The free parameter can be optimized to achieve the optimal fourth-order accuracy. The WCNS scheme with perturbation terms is called the WCNS–PM scheme that is adopted in a smooth stencil. To ensure the numerical stability, a monotone polynomial interpolation method is applied to filter out the non-smooth stencils where the WCNS–ZM scheme is applied. Numerical simulation results indicate the remarkable advantages of the two new schemes in terms of accuracy, resolution and computational efficiency, compared to the classical WCNS–JS and WCNS–Z schemes.</p>