The rapid development of metal 3D printing techniques has enabled the exploration of complex free-convection heat sink designs. Small free-convection heat sinks with pin-fin arrays (or novel geometries) are widely employed at different orientations in a variety of electronic devices, yet there is limited understanding of how orientation impacts their heat transfer behavior. This article characterizes the orientation-dependent performance of a small, tapered pin, free-convection heat sink (named HS17) manufactured with direct metal laser sintering for use with a thermoelectric scalp cryotherapy device for the prevention of chemotherapy-induced alopecia. A validated numerical model and custom-built free-convection test rig were used to investigate the heat sink’s performance over the orientation range of 0 deg to 135 deg. HS17 maintained relatively robust performance over the 0 deg to 90 deg range; however, the thermal resistance (Rth) at 112.5 deg and 135 deg was 6% and 11% higher compared to the 90 deg case, respectively. The heat sink design was modified to include a 22.5 deg wedge base (named HS17-W) to mitigate this performance decline, which is important to ensure safe and continued operation of the cryotherapy device. Compared to the flat base heat sink, the wedge-base design successfully reduced Rth from 11.9 K/W, 12.5 K/W, and 12.8 K/W to 11.5 K/W, 11.8 K/W, and 12.3 K/W at 90 deg, 112.5 deg, and 135 deg, respectively. These results demonstrate the effectiveness of the current proposed design to improve the performance of free-convection heat sinks at downward-facing orientations.