While rerouting existing demands towards a (better) target configuration, it is sometimes unavoidable to end up in an intermediary deadlock state, where some demands need to be rate-limited to be able to successfully reroute all demands to their new path. This is the classic Deadlock Break Problem (DBP), known to be NP-hard. We propose to simplify DBP by first selecting a subset of demands such that, once they have been rerouted via rate-limiting, then all demand rates can be restored to their original value and the normal rerouting process can resume. We call this sub-problem Deadlock Jump Problem (DJP), which is to be solved prior to DBP. We prove that DJP is NP-hard, but we provide an efficient heuristic for it, consisting in rerouting backward the demands in deadlock, i.e., from their target to their initial path. Next, DBP can focus only on the demands that have not been successfully rerouted backward. We finally show by experiments that our approach can significantly boost the performance of State-of-the-Art algorithms for DBP, in terms of throughput gain, number of rate-limited demands and number of rerouting epochs.