Simulations of a leading edge of a hypersonic vehicle using computational fluid dynamics (CFD) and a material response code are presented in order to investigate the effect in-depth surface conduction has on electron transpiration cooling (ETC). ETC is a recently proposed thermal management approach. Previous numerical studies have shown that ETC can significantly lower the stagnation point surface temperature of sharp leading edges of hypersonic vehicles. However, these studies have neglected the effect of heat also being conducted into the material as opposed to only into the flow via radiative cooling and ETC. A modeling approach is presented for ETC, which includes the boundary conditions for electron emission from the surface, accounting for the electric field and space-charge limit effects within the near-wall plasma sheath. A material response code is used to determine typical values of in-depth surface conduction for the test cases studied. Since ETC materials are still being developed, a parametric study is conducted for a range of material properties pertinent to ETC. The results of this study are used to generate in-depth surface conduction profiles, which are implemented into the CFD framework. The CFD simulations show that including in-depth surface conduction results in lower surface temperatures than predicted with radiative and ETC cooling alone. This is because in-depth surface conduction complements radiative cooling and ETC by moving heat away from the surface, in the case of surface conduction by moving the energy into the material, allowing for a lower surface temperature. The results also show that ETC remains a major mode of heat transfer away from the surface, even with in-depth surface conduction. This suggests that ETC is still a promising mode of thermal management, especially since it transfers energy to the flow instead of into the material.