Due to the microscopic roughness of contacting materials, an additional thermal resistance arises from the constriction and spreading of heat near contact spots. Predictive models for contact resistance typically consider abutting semi-infinite cylinders subjected to an adiabatic boundary condition along their outer radius. At the nominal plane of contact an isothermal and circular contact spot is surrounded by an adiabatic annulus and the far-field boundary condition is constant heat flux. However, cylinders with flat bases do not mimic the actual geometry of contacts. To remedy this, we perturb the geometry of the problem such that, in cross section, the circular contact is surrounded by an adiabatic arc. When the curvature of this arc is small, our solution is semi-analytical. Then, we employ a series solution for leading-order (flat-base) problem and use Green's Second Identity to compute the increase in contact resistance without needing to resolve the temperature field. Complimentary numerical results for contact resistance span the full range of geometric parameters, i.e., contact fraction and protrusion angle of the arc. The results suggest as much as a 10-15% increase in contact resistance for realistic contacts for typical contact sizes and asperity slopes.