The development of high-order shock-capturing schemes is critical for compressible fluid simulations, in particular for cases where both shock waves and small-scale turbulence structures present. As one of the state-of-the-art high-order numerical schemes, the family of high-order targeted ENO (TENO) schemes proposed by Fu et al. [Journal of Computational Physics 305 (2016): 333-359] has been demonstrated to perform well for compressible gas dynamics on structured meshes and recently extended to unstructured meshes by Ji et al. [Journal of Scientific Computing 92(2022): 1-39]. In this paper, with the observation that the TENO scheme not only provides the high-order reconstructed data at the cell interface but also features the potential to separate the local flow scales in the wavenumber space, we propose a low-dissipation finite-volume TENO scheme with a new cell-interface flux evaluation strategy for unstructured meshes. The novelty originates from the fact that the local flow scales are classified, following a strong scale separation in the reconstruction process, as “very smooth” or not. When the corresponding large central-biased stencil for the targeted cell interface is judged to be “very smooth”, a low-dissipation Riemann solver, even the non-dissipative central flux scheme, is employed for the cell-interface flux computing. Otherwise, a dissipative approximate Riemann solver is employed to avoid spurious oscillations and achieve stable shock-capturing. Such a strategy provides separate control over the numerical dissipation of the high-order reconstruction process and the cell-interface flux calculation within a unified framework and leads to a resultant finite-volume method with extremely low-dissipation properties and good numerical robustness. Without parameter tuning case by case, a set of canonical benchmark simulations has been conducted to assess the performance of the proposed scheme.