Adding new variable to model
I do not think you can estimate the effect of a variable without adding it to the model. This is because the effect of a variable on the model's discriminatory power depends on 


the strength of association between the outcome variable and the new variable
whether the new variable are collinear with some of the old variables


You could in principle estimate both strength of association and collinearity, but it probably is bad practice and would result in overfitting.

Also in general I think it is best to not use accuracy to evaluate a logistic regression (see ref) but rather a proper scoring rule like Brier Score.

Further, when comparing two nested models (i.e. where one model contains a subset of the variables of the other model) I believe that best practice is to compare the AIC or BIC, or perform a likelihood ratio test.