In this article, we prove some results concerning the truncated two-point function of the infinite-range Ising model above and below the critical temperature. More precisely, if the coupling constants are of the form $J_{x}= ψ(x)e^{ -ρ(x)}$ with $ρ$ some norm and $ψ$ an subexponential correction, we show under appropriate assumptions that given $s\in\mathbb{S}^{d-1}$, the Laplace transform of the two-point function in the direction $s$ is infinite for $β=β_{\text{sat}}(s)$ (where $β_{\text{sat}}(s)$ is a the biggest value such that the inverse correlation length $ν_β(s)$ associated to the truncated two-point function is equal to $ρ(s)$ on $[0,β_{\text{sat}}(s)))$. Moreover, we prove that the two-point function satisfies Ornstein-Zernike asymptotics for $β=β_{\text{sat}}(s)$ on $\mathbb{Z}$. As far as we know, this constitutes the first result on the behaviour of the two-point function at $β_{\text{sat}}(s)$. Finally, we show that there exists $β_{0}$ such that for every $β>β_{0}$, $ν_β(s)=ρ(s)$. All the results are new.