Incredible progress on the theoretical uncertainty of the spatial correlations of the stochastic gravitational wave (GW) background were recently made. However, it remains to realize the impact of this theoretical uncertainty on PTA cross correlations analysis. This paper pushes forward in this direction, as a proof-of-principle: showing the potential role that theoretical uncertainty has on unburying the stochastic GW background signal in noisy PTA cross correlation measurements. We consider both a mock data set and the noise-marginalized 12.5 years NANOGrav spatial correlation measurements, and find optimistic conclusions regardless of the physical content of the GW background and the nature of the noise in the data. Very briefly, we show through various cases a modest result that looking out for a stochastic signal is better when two of its moments are utilized. Or, in terms of GWs, we show that the theoretical uncertainty can play a substantial role in the hunt for the stochastic GW background.