Deflagrations in porous solid propellants are often affected by an increasing pressure difference, or overpressure, between the burned-gas region and the gas deep within the pores of the material. As a result, there appears to be a relatively rapid change in the burning-rate response over a certain range of overpressures in which the sensitivity, or slope, of the propagation speed as a function of overpressure transitions from relatively small to large values. This is often referred to as a transition from "conductive to "convective burning, corresponding to the increased role played by convective gas transport relative to thermal diffusion in determining the propagation speed of the deflagration. In the present work, we consider the analysis of a two-temperature model in which finite-rate interphase heat-transfer effects also play an important role in determining the burning-rate eigenvalue. In particular, we revisit a physically relevant scenario in which the first effects of temperature nonequilibrium are felt in the thin multiphase reaction region, with the preheat zone remaining in thermal equilibrium to a first approximation. Expanding on previous asymptotic results that are further specialized to certain limiting parameter regimes, we consider a combination of analytical and numerical approaches to obtain solutions in the chemical boundary layer, and hence the burning-rate eigenvalue, for a significantly wider range of parameters. In particular, we are able to address a greater range of resistance to interphase heat transfer and thus determine an upper limit beyond which interphase temperature differences are no longer negligible in the preheat region. The main result is that, relative to earlier single-temperature models in which temperature-nonequilibrium effects are completely neglected, the burning-rate response exhibits a much sharper transition from the conduction- to the convection-dominated regime. This results from the ability of the reactive phase to retain a greater amount of the heat of reaction, causing a rapid increase in the reaction rate as the local temperature in that phase exceeds both the corresponding single-temperature value and even the final burned temperature.