A common design for cooling the combustor liner of gas turbines is a double wall composed of impingement jets that feed effusion cooling holes. An important cooling mechanism associated with the effusion hole is the convective cooling provided to the liner wall, which is in contact with the hot main gas flowing through the combustor. While the combination of impingement jets and effusion holes has been studied earlier, mostly in terms of cooling effectiveness, investigators have not fully evaluated the effect the impingement jet has on the local internal convection within the effusion hole. This study evaluates the detailed effects of the impingement geometry on the local convective heat transfer coefficients within the effusion hole, which provides insights as to the design decisions for cooling combustor liners. Using a scaled-up, 3D-printed effusion hole with a constant heat flux boundary condition, local convective heat transfer coefficients were measured for a range of impingement geometries and positions relative to the effusion holes. Results showed a strong influence on the convective heat transfer resulting from the placement of the impingement hole relative to the effusion hole. In particular, the results showed a strong sensitivity to the circumferential and radial placement of the impingement jet with little sensitivity to the jet-to-effusion distance.