We study the zero-free regions of the partition function of the hard-core model on finite graphs and their implications for the analyticity of the free energy on infinite lattices. Classically, zero-freeness results have been established up to the tree uniqueness threshold $λ_c(Δ-1)$ determined by the maximum degree $Δ$. However, for many graph classes, such as regular lattices, the connective constant $σ$ provides a more precise measure of structural complexity than the maximum degree. While recent approximation algorithms based on correlation decay and Markov chain Monte Carlo have successfully exploited the connective constant to improve the threshold to $λ_c(σ)$, analogous results for complex zero-freeness have been lacking. In this paper, we bridge this gap by introducing a proper definition of the connective constant for finite graphs based on a lower bound on the number of $k$-depth self-avoiding walks. We prove that for any graph family with a lower connective constant $μ$, the partition function is zero-free in a complex neighborhood of the interval $[0, λ]$ for all $λ< λ_c(μ)$. As a direct consequence, we establish the uniqueness and analyticity of the free energy density for infinite lattices up to the connective constant threshold, extending the known regions derived from maximum degree bounds. Our proof utilizes a block contraction technique that lifts the correlation decay property from a real interval to a strip-like complex neighborhood.