How to improve precision under imbalanced classification
Introducing a Strong Feature would definitely help as it is "Strong" :-) .. if you do not have a sure-success feature then you may try changing penalty of miss-classification to start with. 

You may try synthetic (i.e. SMOTE) or non-synthetic (domain based) approaches to bulk up the lower class.

Also, if this is a very very rare class the repeated sampling techniques may work