This article addresses the concern regarding late-onset dose-limiting toxicities (DLT), moderate toxicities below the threshold of a DLT and cumulative toxicities that may lead to a DLT, which are mostly disregarded or handled in an ad hoc manner when determining the maximum tolerated dose (MTD) in dose-finding cancer clinical trials. An extension of the Time-to-Event Continual Reassessment Method (TITE-CRM) which allows for the specification of toxicity constraints on both DLT and moderate toxicities, and can account for partial information is proposed. The method is illustrated in the context of an Erlotinib dose-finding trial with low DLT rates, but a significant number of moderate toxicities leading to treatment discontinuation in later cycles. Based on simulations, our method performs well at selecting the dose level that satisfies both the DLT and moderate-toxicity constraints. Moreover, it has similar probability of correct selection compared to the TITE-CRM when the true MTD based on DLT only and the true MTD based on grade 2 or higher toxicities alone coincide, but reduces the probability of recommending a dose above the MTD.