Stashing only un-staged changes in Git

git stash push has an option --keep-index that does exactly what you need.

So, run git stash push --keep-index.

This may be done in 3 steps: save staged changes, stash everything else, restore index with staged changes. Which is basically:

git commit -m 'Save index'
git stash push -u -m 'Unstaged changes and untracked files'
git reset --soft HEAD^

This will do exactly what you want.

git stash save --keep-index

Also, Re:

Why not commit your changes after staging them? – Shin

A: Because you should always checkin tested code :) That means, you need to run the tests with only the changes you are about to commit

_{All this apart from the fact that of course, as an experienced programmer, you have the innate urge to test and review just those changes -- only partly kidding}