The integration of three-dimensional microelectronics is hampered by overheating issues inherent to state-of-the-art integrated circuits. Fundamental understanding of heat transfer across soft-solid interfaces is important for developing efficient heat dissipation capabilities. At the microscopic scale, the formation of a dense liquid layer at the solid-liquid interface decreases the interfacial heat resistance. We show through molecular dynamics simulations of n-perfluorohexane on a generic wettable surface that enhancement of the liquid structure beyond a single adsorbed layer drastically enhances interfacial heat conductance. Pressure is used to control the extent of the liquid layer structure. The interfacial thermal conductance increases with pressure values up to 16.2 MPa at room temperature. Furthermore, it is shown that liquid structuring enhances the heat-transfer rate of high-energy lattice waves by broadening the transmission peaks in the heat flux spectrum. Our results show that pressure is an important external parameter that may be used to control interfacial heat conductance at solid-soft interfaces.