We extend previous work on gamma-ray burst afterglows involving hot thermal electrons at the base of a shock-accelerated tail. Using a physically motivated electron distribution based on first-principles simulations, we compute the broadband emission from radio to TeV gamma rays. For the first time, we present the effects of a thermal distribution of electrons on synchrotron self-Compton emission. The presence of thermal electrons causes temporal and spectral structure across the entire observable afterglow, which is substantively different from models that assume a pure power-law distribution for the electrons. We show that early-time TeV emission is enhanced by more than an order of magnitude for our fiducial parameters, with a time-varying spectral index that does not occur for a pure power law of electrons. We further show that the X-ray closure relations take a very different, also time-dependent, form when thermal electrons are present; the shape traced out by the X-ray afterglows is a qualitative match to observations of the traditional decay phase.