Discrete-time survival analysis (DTSA) models are a popular way of modeling events in the social sciences. However, the analysis of discrete-time survival data is challenged by missing data in one or more covariates. Negative consequences of missing covariate data include efficiency losses and possible bias. A popular approach to circumventing these consequences is multiple imputation (MI). In MI, it is crucial to include outcome information in the imputation models. As there is little guidance on how to incorporate the observed outcome information into the imputation model of missing covariates in DTSA, we explore different existing approaches using fully conditional specification (FCS) MI and substantive-model compatible (SMC)-FCS MI. We extend SMC-FCS for DTSA and provide an implementation in the smcfcs R package. We compare the approaches using Monte Carlo simulations and demonstrate a good performance of the new approach compared to existing approaches.