We study the problem of enumeration of all k-sized subsets of temporal events that mutually overlap at some point in a query time window. This problem arises in many application domains, e.g., in social networks, life sciences, smart cities, telecommunications, and others. We propose a start time index (STI) approach that overcomes the efficiency bottlenecks of current methods which are based on 2-way join algorithms to enumerate temporal k-cliques. Additionally, we investigate how precomputed checkpoints can be used to further improve the efficiency of STI. Our experimental results demonstrate that STI outperforms the state of the art by a wide margin and that our checkpointing strategies are effective.