We investigate the correlation between spin signals measured in three-terminal (3T) geometry by the Hanle effect and the spin accumulation generated in a semiconductor channel in a lateral (Ga,Mn)As/GaAs Esaki diode device. We systematically compare measurements using a 3T configuration, probing spin accumulation directly beneath the injecting contact, with results from nonlocal measurements, where solely spin accumulation in the GaAs channel is probed. We find that the spin signal detected in the 3T configuration is dominated by a bias-dependent spin detection sensitivity, which in turn is strongly correlated with charge-transport properties of the junction. This results in a particularly strong enhancement of the detected spin signal in a region of increased differential resistance. We find additionally that two-step tunneling via localized states in the gap of (Ga,Mn)As does not compromise spin injection into the semiconductor conduction band.