In the Sequential Selection Problem (SSP), immediate and irrevocable decisions need to be made as candidates randomly arrive for a job interview. Standard SSP variants, such as the well-known secretary problem, begin with an empty selection set (cold-start) and perform the selection process once over a single candidate set (single-round). In this paper we address these two limitations. First, we introduce the novel Warm-starting SSP (WSSP) setting which considers at hand a reference set, a set of previously selected items of a given quality, and tries to update optimally that set by (re-)assigning each job at most once. We adopt a cutoff-based approach to optimize a rank-based objective function over the final assignment of the jobs. In our technical contribution, we provide analytical results regarding the proposed WSSP setting, we introduce the algorithm Cutoff-based Cost Minimization (CCM) (and the low failures-CCM, which is more robust to high rate of resignations) that adapts to changes in the quality of the reference set thanks to the translation method we propose. Finally, we implement and test CCM in a multi-round setting that is particularly interesting for real-world application scenarios.