We continue the study of renewal contact processes initiated in a companion paper, where we showed that if the tail of the interarrival distribution $μ$ is heavier than $t^{-α}$ for some $α<1$ (plus auxiliary regularity conditions) then the critical value vanishes. In this paper we show that if $μ$ has decreasing hazard rate and tail bounded by $t^{-α}$ with $α>1$, then the critical value is positive in the one-dimensional case. A more robust and much simpler argument shows that the critical value is positive in any dimension whenever the interarrival distribution has a finite second moment.