The renewal contact process, introduced in $2019$ by Fontes, Marchetti, Mountford, and Vares, extends the Harris contact process in $\mathbb{Z}^d$ by allowing the possible cure times to be determined according to independent renewal processes (with some interarrival distribution $μ$) and keeping the transmission times determined according to independent exponential times with a fixed rate $λ$. We investigate sufficient conditions on $μ$ to have a process with a finite critical value $λ_c$ for any spatial dimension $d \geq 1$. In particular, we show that $λ_c$ is finite when $μ$ is continuous with bounded support or when $μ$ is absolutely continuous and has a decreasing hazard rate.