I'm no expert but it sounds like you want to perform logistic regression with 'successful at stage 2' as your dependent variable.

You then have predictor variables, one of which is the binary variable 'successful at stage 1', plus your other predictor variables.

You'll then see if any of the predictor variables (including successful at stage 1) are significant predictors of successful at stage 2.

Given your small sample size, however, you can't really have too many predictors in the model. So a backwards or forwards regression strategy where the software throws out predictors or adds them in depending on their significance would probably work best.