Does a precision score increasing with a higher number of folds mean the model will improve with more data?
It may or may not indicate that more data will help.

The correct way to see if model performance increases with more data is to create a learning curve. Train your model on a small subset of your data with cross-validation, and then keep increasing the amount of data used. This is not just reducing/increasing the number of folds. It is actually limiting yourself to a small amount of data for your 10-fold cross-validation and increasing it, evaluating the performance at each increase. Here's more about a learning curve.