The logistic regression model is a direct probability estimation method. Classification should play no role in its use. Any classification not based on assessing utilities (loss/cost function) on individual subjects is inappropriate except in very special emergencies. The ROC curve is not helpful here; neither are sensitivity or specificity which, like overall classification accuracy, are improper accuracy scoring rules that are optimized by a bogus model not fitted by maximum likelihood estimation.
Note that you achieve high predictive discrimination (high $c$-index (ROC area)) by overfitting the data. You need perhaps at least $15p$ observations in the least frequent category of $Y$, where $p$ is the number of candidate predictors being considered, in order to obtain a model that is not significantly overfitted [i.e., a model that is likely to work on new data about as well as it worked on the training data]. You need at least 96 observations just to estimate the intercept such that the predicted risk has a margin of error $\leq 0.05$ with 0.95 confidence.