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 Jx = ψ(x)e^-ρ(x) with ρ some norm and ψ an subexponential correction, we show under appropriate assumptions that given s ∈ S^d-1, the Laplace transform of the two-point function in the direction s is infinite for β = β_sat(s) (where β_sat(s) is a the biggest value such that the inverse correlation length νβ(s) associated to the truncated two-point function is equal to ρ(s) on [0, β_sat(s))). Moreover, we prove that the two-point function satisfies up-to-constants Ornstein-Zernike asymptotics for β = β_sat(s) on Z. As far as we know, this constitutes the first result on the behaviour of the two-point function at β_sat(s). Finally, we show that there exists β₀ such that for every β > β₀, ν_β(s) = ρ(s). All the results are new and their proofs are built on different results and ideas developed in Duminil-Copin and Tassion (Commun Math Phys 359(2):821–822, 2018) and Aoun et al. in (Commun Math Phys 386:433–467, 2021).