Curiosity states benefit memory for target information, but also incidental information presented during curiosity states. However, it is not known whether incidental curiosity-enhanced memory depends on when incidental information during curiosity states is encountered. Here, participants incidentally encoded unrelated face images at different time points while they anticipated answers to trivia questions. Across two experiments, we found memory enhancements for unrelated faces presented during high-curiosity compared with low-curiosity states, but only when presented shortly after a trivia question. This suggests processes associated with the elicitation of curiosity--but not sustained anticipation or the satisfaction of curiosity--enhance memory for incidental information.