We study the contact process with stirring on $\mathbb{Z}^d$. In this process, particles occupy vertices of $\mathbb{Z}^d$; each particle dies with rate 1 and generates a new particle at a randomly chosen neighboring vertex with rate $λ$, provided the chosen vertex is empty. Additionally, particles move according to a symmetric exclusion process with rate $N$. For any $d$ and $N$, there exists $λ_c$ such that, when the system starts from a single particle, particles go extinct when $λ< λ_c$ and have a chance of being present for all times when $λ> λ_c$. Durrett and Neuhauser proved that $λ_c$ converges to 1 as $N$ goes to infinity, and Konno, Katori and Berezin and Mytnik obtained dimension-dependent asymptotics for this convergence, which are sharp in dimensions 3 and higher. We obtain a lower bound which is new in dimension 2 and also gives the sharp asymptotics in dimensions 3 and higher. Our proof involves an estimate for two-type renewal processes which is of independent interest.