A well-known approach to optimally solving Multi-Agent Path Finding (MAPF) is by compilation to Boolean Satisfiability or Answer Set Programming. Such compilation-based approaches to MAPF are superior to others on dense, relatively small instances. During solving, the underlying solver is invoked multiple times, each with an encoding of the same instance for a different makespan. The runtime of the last solver invocation, whose input is the instance encoded with a theoretical upper bound of the makespan of the optimal solution, is critical to performance. This paper proposes a new theoretical upper bound for such a last invocation, which we prove is correct. Unlike the previously known bound, given a MAPF instance, our bound requires computing a semi-relaxed solution, which is the union of cost-optimal solutions for partitions of such an instance. The computation of our new bound requires optimally solving partitions, which requires more computational resources than those needed for computing the bound currently used by state-of-the-art solvers. We propose a recursive parallel approach that, we call ReBo, which despite additional overhead in upper bound computation, obtains substantially better overall results by exploiting the new bound. ReBo uses a heuristic to select an appropriate partition for bound computation, which does not guarantee that the solution returned is optimal. However, in our benchmarks, composed of 2,890 problems over warehouses and random instances, ReBo never finds a suboptimal solution. In addition, we found that the new bound is significantly tighter than the previously known, on average, 21.2% smaller than the previous bound. This allows us to generate encodings that are around 33.45% smaller, allowing ReBo to solve 4.9% more instances than a state-of-the-art solver in our benchmark set.