We re-evaluate the non-perturbative contribution to the thrust distribution in $e+e-\to$ hadrons, in the light of the latest experimental data and the recent NNLO perturbative calculation of this quantity. By extending the calculation to NNLO+NLL accuracy, we perform the most detailed study to date of the effects of non-perturbative physics on this observable. In particular, we investigate how well a model based on a low-scale QCD effective coupling can account for such effects. We find that the difference between the improved perturbative distribution and the experimental data is consistent with a $1/Q$-dependent non-perturbative shift in the distribution, as predicted by the effective coupling model. Best fit values of $\alpha s(MZ)=0.1164-0.0026+0.0028$ and $\alpha 0(2GeV)=0.59+/-0.03$ are obtained with $\chi 2/d.o.f.=1.09$. This is consistent with NLO+NLL results but the quality of fit is improved. The agreement in $\alpha 0$ is non-trivial because a part of the 1/Q-dependent contribution (the infrared renormalon) is included in the NNLO perturbative correction.