We continue the study of renewal contact processes initiated in a companion paper, where we showed that if the tail of the interarrival distribution mu is heavier than t(-alpha) for some alpha < 1 (plus auxiliary regularity conditions) then the critical value vanishes. In this paper we show that if mu has decreasing hazard rate and tail bounded by t(-alpha) with alpha > 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. (C) 2019 Published by Elsevier B.V.