The asymptotic behavior of the integrated density of states (IDS), \(N(E)\), is investigated for random Schrödinger operators with a single-site potential \(V\) satisfying \(\mathrm{essinf}\, V = -\infty\). Under the assumption that the underlying point process is a Gibbs point process with repulsive pairwise interactions, the leading term of \(\log N(E)\) as \(E \to -\infty\) is determined using a periodic approximation method. It is shown that repulsive pairwise interactions lead to a significantly faster decay of \(N(E)\) compared to the Poisson case. Furthermore, configurations with multiple clusters can provide the dominant contribution to the IDS in the Gibbs setting, contrasting with the single-cluster dominance typically observed in Poisson models. Finally, refined estimates of the leading constants are provided for specific classes of potentials, including those with multiple singularities.