Gas turbine cooling system design is constrained by a maximum allowable wall temperature (dictated by the material, the life requirements of the component, and a given stress distribution), the desire to minimize coolant mass flow rate (requirement to minimize cycle-efficiency cost), and the requirement to achieve as close to uniform wall temperature as possible (to reduce thermal gradients, and stress). These three design requirements form the basis of an iterative design process. The relationship between the requirements has received little discussion in the literature, despite being of interest from both a theoretical and a practical viewpoint. In Part I, we show analytically that the coolant mass flow rate is minimized when the wall temperature is uniform and equal to the maximum allowable wall temperature. In this paper, we show that designs optimized for uniform wall temperature have a corresponding optimum internal heat transfer coefficient (HTC) distribution. In this paper, analytical expressions for the optimum internal HTC distribution are derived for a number of cooling systems, with and without thermal barrier coating (TBC). Most cooling systems can be modeled as a combination of these representative systems. The optimum internal HTC distribution is evaluated for a number of engine-realistic systems: long plate systems (e.g., combustors, afterburners), the suction-side (SS) of a high pressure nozzle guide vane (HPNGV), and a radial serpentine cooling passage. For some systems, a uniform wall temperature is unachievable; the coolant penalty associated with this temperature nonuniformity is estimated. A framework for predicting the optimum internal HTC for systems with any distribution of external HTC, wall properties, and film effectiveness is outlined.