We report a new determination of qˆ, the jet transport coefficient of the quark-gluon plasma. We use the JETSCAPE framework, which incorporates a novel multistage theoretical approach to in-medium jet evolution and Bayesian inference for parameter extraction. The calculations, based on the MATTER and LBT jet quenching models, are compared to experimental measurements of inclusive hadron suppression in Au + Au collisions at the BNL Relativistic Heavy Ion Collider (RHIC) and Pb + Pb collisions at the CERN Large Hadron Collider (LHC). The correlation of experimental systematic uncertainties is accounted for in the parameter extraction. The functional dependence of qˆ on jet energy or virtuality and medium temperature is based on a perturbative picture of in-medium scattering, with components reflecting the different regimes of applicability of MATTER and LBT. In the multistage approach, the switch between MATTER and LBT is governed by a virtuality scale Q 0 .Comparison of the posterior model predictions to the RHIC and LHC hadron suppression data shows reasonable agreement, with moderate tension in limited regions of phase space. The distribution of qˆ/T 3 extracted from the posterior distributions exhibits weak dependence on jet momentum and medium temperature T , with 90% credible region (CR) depending on the specific choice of model configuration. The choice of MATTER+LBT, with switching at virtuality Q 0 , has 90% CR of 2 < qˆ/T 3 < 4for p T,jet > 40 GeV/c. The value of Q 0 , determined here for the first time, is in the range 2.0-2.7 GeV.