  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
$e\Delta_{\phi}(\beta)^{-1}$, where
$\Delta_{\phi}(\beta)\in(0,C_{\phi}(\beta)]$ 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
$e^{2-2W(eA_{\phi}(\beta)/\Delta_{\phi}(\beta))}\Delta_{\phi}(\beta)^{-1}e^{-(\beta
C+1)}$, where $C\ge0$ is the local stability constant, $W(\cdot)$ the Lambert
$W$-function, $A_{\phi}(\beta)$ the contribution from the attraction in the
pair potential to the temperedness constant, and
$\Delta_{\phi}(\beta)\in[A_{\phi}(\beta),C_{\phi}(\beta)]$ 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 $C_{\phi}(\beta)^{-1}e^{-(\beta C+1)}$ by at
least a factor of $e^{2}$.
