The architecture of supercomputers is evolving to expose massive parallelism. MPI and OpenMP are widely used in application codes on the largest supercomputers in the world. The community primarily focused on composing MPI with OpenMP before its version 3.0 introduced task-based programming. Recent advances in OpenMP task model and its interoperability with MPI enabled fine model composition and seamless support for asynchrony. Yet, OpenMP tasking overheads limit the gain of task-based applications over their historical loop parallelization (parallel for construct). This paper identifies the OpenMP task dependency graph discovery speed as a limiting factor in the performance of task-based applications. We study its impact on intra and inter-node performances over two benchmarks (Cholesky, HPCG) and a proxy-application (LULESH). We evaluate the performance impacts of several discovery optimizations, and introduce a persistent task dependency graph reducing overheads by a factor up to 15 at run-time. We measure 2x speedup over parallel for versions weak scaled to 16K cores, due to improved cache memory use and communication overlap, enabled by task refinement and depth-first scheduling.