Sensitivity analysis for the unconfoundedness assumption is crucial in observational studies. For this purpose, the marginal sensitivity model gained popularity recently due to good interpretability and mathematical properties. However, most existing models only consider a worst-case parameter that bounds the logit difference between the observed and full data propensity scores, which may not fully capture the extent of unmeasured confounding. We propose a new sensitivity model that is parameterized by the second moment of the propensity score ratio, requiring only the average strength of unmeasured confounding to be bounded. By characterizing the associated sensitivity analysis as an optimization problem, we derive sharp closed-form bounds of the average potential outcomes under our model. We propose efficient one-step estimators for these bounds based on the corresponding efficient influence functions. Additionally, we apply multiplier bootstrap to construct simultaneous confidence bands to cover the sensitivity curve that consists of bounds at different values of the sensitivity parameters. Through a real-data study, we illustrate how this average-case sensitivity analysis can provide tighter bounds and facilitate calibration of the results using observed covariates.