In the family of the iron-based superconductors, the REFeAsO-type compounds (with RE being a rare-earth metal) exhibit the highest bulk superconducting transition temperatures (T-c) up to 55 K and thus hold the key to the elusive pairing mechanism. Recently, it has been demonstrated that the intrinsic electronic structure of SmFe0.92Co0.08AsO (T-c = 18 K) is highly nontrivial and consists of multiple band-edge singularities in close proximity to the Fermi level. However, it remains unclear whether these singularities are generic to the REFeAsO-type materials and if so, whether their exact topology is responsible for the aforementioned record Tc. In this work, we use angle-resolved photoemission spectroscopy (ARPES) to investigate the inherent electronic structure of the NdFeAsO0.6F0.4 compound with a twice higher T-c = 38 K. We find a similarly singular Fermi surface and further demonstrate that the dramatic enhancement of superconductivity in this compound correlates closely with the fine-tuning of one of the band-edge singularities to within a fraction of the superconducting energy gap. below the Fermi level. Our results provide compelling evidence that the band-structure singularities near the Fermi level in the iron-based superconductors must be explicitly accounted for in any attempt to understand the mechanism of superconducting pairing in these materials.