Alternatives for logistic regression?

Hey everyone!

I need some help because I am not really happy with the data analysis for my thesis.

I am investigating the prognostic power of 3 quantitative and 2 qualitative variable on a dichotomous outcome. First off I wanted to know which set of variables is the best to predict the outcome.

Due to the small sample size in my study I couldn't enter all variables at once into a logistic regression, that just produced a lot of error reports. So I conducted a Chi²-test for each of the 2 qualitative variables and the outcome, those results were not significant.
For the 3 quantitative variables I conducted single logistic regression and after the result that all of them were significant, I tried different scenarios stepwise to find the best set of variables to predict the outcome.

So after all I have an result but I am not happy with that since the data analysis seems so disrupted. Is there a smarter way to conduct the analysis, that in best case includes all variables at once?