While the recently proposed TENO (targeted essentially non-oscillatory) schemes [Fu et al., Journal of Computational Physics 305 (2016): 333-359] exhibit better performance than the classical WENO (weighted essentially non-oscillatory) schemes with the same accuracy order, there is still a room for further improvement, e.g., the physical discontinuities may be significantly smeared by the excessive numerical dissipation due to the enforcement of the ENO property after a long-time advection. More recently, a new fifth-order TENO5-THINC scheme is proposed by coupling the TENO5 scheme with a non-polynomial THINC (tangent of hyperbola for interface capturing) scheme based on a parameter-free discontinuity indicator. The novelty originates from the fact that the new strategy locates the discontinuities accurately and deploys the jump-like THINC reconstruction scheme for resolving the discontinuities with a sub-cell resolution, instead of enforcing the ENO property. The new scheme successfully leverages the excellent wave-resolution property of standard TENO schemes for smooth and under-resolved continuous scales and the discontinuity-resolving capability of THINC for reconstructing genuine discontinuities. In this work, we further develop the low-dissipation discontinuity-resolving very-high-order TENO-THINC reconstruction schemes for hyperbolic conservation laws by proposing tailored coupling strategies. Without loss of generality, the six- and eight-point TENO-THINC schemes are developed, and the explicit formulas are given as well as the built-in parameters. Based on a set of critical benchmark simulations, the newly proposed schemes show significantly lower numerical dissipation when compared to the counterpart TENO schemes without sacrificing numerical robustness. The presented numerical results represent the state-of-the-art in the literature and can serve as references for future algorithm development.