We use fully self-consistent GW calculations on diamond and silicon carbide to reparametrize the Heyd-Scuseria-Ernzerhof (HSE) exact exchange density functional for use in band structure calculations of semiconductors and insulators. We show that the thus modified functional is able to calculate the band structure of bulk Si, Ge, GaAs, and CdTe with good quantitative accuracy at a significantly reduced computational cost as compared to GW methods, and also gives significantly improved band gap predictions in wide-gap ionic crystals as compared to the HSE06 parametrization. We discuss the limitations of this functional in low dimensions by calculating the band structures of single-layer hexagonal BN and MoS2, and by demonstrating that the diameter scaling of curvature induced band gaps in single-walled carbon nanotubes is still physically incorrect using our functional; we consider possible remedies to this problem.