  We report a new determination of $\hat{q}$, the jet transport coefficient of
the Quark-Gluon Plasma. We use the JETSCAPE framework, which incorporates a
novel multi-stage 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 RHIC and Pb+Pb collisions
at the LHC. The correlation of experimental systematic uncertainties is
accounted for in the parameter extraction. The functional dependence of
$\hat{q}$ 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 multi-stage
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 $\hat{q}/T^3$ 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 $2<\hat{q}/T^3<4$ for
$p_\mathrm{T}^\mathrm{jet}>40$ GeV/c. The value of $Q_0$, determined here for
the first time, is in the range 2.0-2.7 GeV.
