In their recent works [Comm. Math. Phys. 399:1 (2023)] and [arXiv:2109.01094], Michelen and Perkins proved that the pressure of a system of particles with repulsive pair interactions is analytic for activities up to $eΔ_φ(β)^{-1}$, where $Δ_φ(β)\in(0,C_φ(β)]$ is a constant they called the potential-weighted connective constant. This paper extends their method to locally stable, tempered, and hard-core pair potentials. Our main result is that the pressure of such a system is analytic for activities up to $e^{2-2W(eA_φ(β)/Δ_φ(β))}Δ_φ(β)^{-1}e^{-(βC+1)}$, where $C\ge0$ is the local stability constant, $W(\cdot)$ the Lambert $W$-function, $A_φ(β)$ the contribution from the attraction in the pair potential to the temperedness constant, and $Δ_φ(β)\in[A_φ(β),C_φ(β)]$ a counterpart of the constant defined by Michelen and Perkins. The main ingredients in the proof include a recursive identity for the one-point density tailored to locally stable hard-core potentials and a corresponding notion of modulations of an activity function. In the high-temperature regime, our result surpasses the classical Penrose-Ruelle bound of $C_φ(β)^{-1}e^{-(βC+1)}$ by at least a factor of $e^{2}$.