Accurately estimating the direction-of-arrival (DOA) of a speech source using a compact microphone array (CMA) is often complicated by background noise and reverberation. A commonly used DOA estimation method is the steered response power with phase transform (SRP-PHAT) function, which has been shown to work reliably in moderate levels of noise and reverberation. Since for closely spaced microphones the spatial coherence of noise and reverberation may be high over an extended frequency range, this may negatively affect the SRP-PHAT spectra, resulting in DOA estimation errors. Assuming the availability of an auxiliary microphone at an unknown position which is spatially separated from the CMA, in this paper we propose to compute the SRP-PHAT spectra between the microphones of the CMA based on the SRP-PHAT spectra between the auxiliary microphone and the microphones of the CMA. For different levels of noise and reverberation, we show how far the auxiliary microphone needs to be spatially separated from the CMA for the auxiliary microphone-based SRP-PHAT spectra to be more reliable than the SRP-PHAT spectra without the auxiliary microphone. These findings are validated based on simulated microphone signals for several auxiliary microphone positions and two different noise and reverberation conditions.