We consider the Gibbs measure of a general interacting particle system for a certain class of ``weakly interacting" kernels. In particular, we show that the local point process converges to a Poisson point process as long as the inverse temperature $β$ satisfies $N^{-1} \ll β\ll N^{-\frac{1}{2}}$, where $N$ is the number of particles. This expands the temperature regime for which convergence to a Poisson point process has been proved.