This study numerically investigates a phase-change material heat sink using gallium and Primitive Triply Periodic Minimal Surface (TPMS) cells structure as in-fill for applications requiring high heat-flux dissipation, e.g., electronics. Initially, we used only gallium without thermal conductivity enhancers, relying solely on buoyancy-driven circulations of melted gallium for the convective cooling of a hot sink base. However, this study aims to surpass gallium’s innate heat-dissipation capability. Therefore, the sink is enhanced with a high conductivity Primitive TPMS cells structure. Hybridization of conduction and convection heat dissipation enhancements is accomplished by using only one layer of the TPMS cells installed at the hot sink base. Comparison with the full TPMS structure filling option is made. In the full TPMS structure filling option, the sink relies mainly on conduction through the TPMS structure’s body and less on natural convection. However, equipping the sink with TPMS structures reduces the overall latent heat storage capacity. Therefore, additional gallium is added to the sink to match the overall gallium content in the baseline case without TPMS structures. Furthermore, the effect of the boundary conditions applied at the wall is explored. The results show considerable improvements in heat dissipation and peak temperatures with the TPMS cells, where by the time 100 s (about 20 s post complete melting) peak temperatures are about 13 percent lower than in the case without TPMS structures. Furthermore, the full structure filling option demonstrates superior performance over the partially filled ones, highlighting the importance of conduction through TPMS cells over convection in the free gallium space. Unlike the partial filling option, conduction in the full filling option has a continuous favorable impact on heat dissipation from the onset of the heating process.