When analyzing time-to-event data, it often happens that some subjects do not experience the event of interest. Survival models that take this feature into account (called `cure models') have been developed in the presence of covariates. However, the current literature on nonparametric cure models with covariates cannot be applied when the follow-up is insufficient, i.e., when the right endpoint of the support of the censoring time is strictly smaller than that of the survival time of the susceptible subjects. In this paper we attempt to fill this gap in the literature by proposing new estimators of the conditional cure rate and the conditional survival function using extrapolation techniques coming from extreme value theory. We establish the asymptotic normality of the proposed estimators, and show how the estimators work for small samples by means of a simulation study. We also illustrate their practical applicability through the analysis of data on the survival of colon cancer patients.