A stepped-bar apparatus has been designed and constructed to characterize the thermal resistance of materials using steady-state heat transfer techniques. The design of the apparatus is a modification of the ASTM D5470 standard where reference bars of equal cross-sectional area are used to extrapolate surface temperatures and heat flux across a sample of unknown thermal resistance. The design modification involves intentionally oversizing the upper reference bar (URB) of the apparatus to avoid contact area uncertainty due to reference bar misalignment, which is difficult to account for, as well as the high cost that can be associated with equipping the apparatus with precise alignment controls (e.g., pneumatic alignment). Multidimensional heat transfer in the upper reference bar near the sample interface is anticipated using numerical modeling. The resulting nonlinear temperature profile in the upper reference bar is accounted for by fitting a second order regression line through thermocouple readings near the sample interface. The thermal resistances of commercially available thermal gap pads and thermal pastes were measured with the stepped-bar apparatus; the measured values were in good agreement with published results, and exhibited a high degree of reproducibility. The measurement uncertainty of both the standard and stepped-bar apparatus decrease with increased thermocouple precision. Notably, the uncertainty due to reference bar misalignment with the standard apparatus becomes more pronounced as thermocouple precision and the number of thermocouples increases, which suggests that the stepped-bar apparatus would be especially advantageous for enabling accurate, high-precision measurements of very low thermal resistances.