Alternatives to Logistic Regression
Any classifier that naturally provides feature importance can be applied here. For example, take a look at Decision Trees and Random Forrest.

For other algorithms that do not provide such information (eg KNN and Neural Nets), you may take a (forward/backward/bidirectional) feature selection. For example, chose a subset of features, train a model and evaluate it with any preferred metrics (better to use bootstrapping or cross-validation). Choose another subset if features and try again. After several trials, you can find the subset with the best prediction power. This is the set of the mist contributing features. 

Note that different techniques/measures may give you different results.