A salient feature of the post-COVID inflation surge is that economic activity has remained resilient despite unfavorable supply-side developments. We develop a macroeconomic model with nonlinear price and wage Phillips curves, endogenous intrinsic indexation and an unobserved components representation of a cost-push shock that is consistent with these observations. In our model, a persistent large adverse supply shock can lead to a persistent inflation surge while output expands if the central bank follows an inflation forecast-based policy rule and thus abstains from hiking policy rates for some time as it (erroneously) expects inflationary pressures to dissipate quickly. A standard linearized formulation of our model cannot account for these observations under identical assumptions. Our nonlinear framework implies that the standard prescription of "looking through" supply shocks is a good policy for small shocks when inflation is near the central bank's target, but that such a policy may be quite risky when economic activity is strong and large shocks drive inflation well above target. Moreover, our model implies that the economic costs of "going the last mile" – i.e. a tight stance aimed at returning inflation quickly to target – can be substantial.