  Signatures of self-organized criticality (SOC) have recently been observed in
an ultracold atomic gas under continuous laser excitation to
strongly-interacting Rydberg states [S. Helmrich et al., Nature, 577, 481--486
(2020)]. This creates a unique possibility to study this intriguing dynamical
phenomenon, e.g., to probe its robustness and universality, under controlled
experimental conditions. Here we examine the self-organizing dynamics of a
driven ultracold gas and identify an unanticipated feedback mechanism, which is
especially important for systems coupled to thermal baths. It sustains an
extended critical region in the trap center for a notably long time via
hydrodynamic transport of particles from the flanks of the cloud toward the
center. This compensates the avalanche-induced atom loss and leads to a
characteristic flat-top density profile, providing an additional experimental
signature for SOC and minimizing effects of inhomogeneity on the SOC features.
