High fidelity numerical simulation of compressible flow requires the numerical method being used to have both stable shock-capturing capability and high spectral resolution. Recently, a family of Targeted Essentially Non-Oscillatory (TENO) schemes is developed to fulfill such requirements. Although TENO has very low dissipation for smooth flow, it introduces a cutoff value CT to maintain the non-oscillatory shock-capturing property. As CT actually controls the dissipation property of TENO, the choice of CT for better shock capturing capability also means higher dissipation for small structures. To overcome this, in this paper, a new local adaptive method is proposed for the choice of CT. By introducing a novel adaptive function based on the WENO smoothness indicators, CT is dynamically adjusted from 1.0 x 10(-10) for lower dissipation to 1.0 x 10(-4) for stable capturing of shock according to the smoothness of the reconstruction stencil. The numerical results of the new method are compared with those of the original TENO method and an adaptive TENO method in Fu et al. (2019) [49]. It reveals that the new method is capable of suppressing numerical oscillations near discontinuities while further improving the resolution of TENO at a low extra computational cost. (c) 2020 Elsevier Inc. All rights reserved.