Schottky barrier height (SBH) engineering of contact structures is a primary challenge to achieve high performance in nanoelectronic and optoelectronic applications. Although SBH can be lowered through various Fermi-level (FL) unpinning techniques, such as a metal/interlayer/semiconductor (MIS) structure, the room for contact metal adoption is too narrow because the work function of contact metals should be near the conduction band edge (CBE) of the semiconductor to achieve low SBH. Here, we propose a novel structure, the metal/transition metal dichalcogenide/semiconductor structure, as a contact structure that can effectively lower the SBH with wide room for contact metal adoption. A perpendicularly integrated molybdenum disulfide (MoS2) interlayer effectively alleviates FL pinning by reducing metal-induced gap states at the MoS2/semiconductor interface. Additionally, it can induce strong FL pinning of contact metals near its CBE at the metal/MoS2 interface. The technique using FL pinning and unpinning at metal/MoS2/semiconductor interfaces is first introduced in the MIS scheme to allow the use of various contact metals. Consequently, significant reductions of the SBH from 0.48 to 0.12 eV for GaAs and from 0.56 to 0.10 eV for Ge are achieved with several different contact metals. This work significantly reduces the dependence on contact metals with lowest SBH and proposes a new way of overcoming current severe contact issues for future nanoelectronic and optoelectronic applications.