The CSS file that sets the quicknavigator position is located at 

yoursite\appdata\Modules\CMS\Util\styles\quicknavigator.css

or just override #epi-quickNavigator with your own styles.

I can't override the CSS that I want to change (unless I want to do it in a ugly hacky way), since quicknavigator.css is injected as the last css file.

For some reason there is no "Util" directory under AppData/Modules/CMS.

I tried searching for it earlier but Windows couldn't find quicknavigator.css in any AppData directory.

My bad, I forgot to ask which EPiServer version. My answer is valid for 7.5. The advantage of the new appdata approach is that changes to the "core" files will only affect that particular site, not all sites on your machine/server.

For earlier 7.x versions, you'll find the file under Program Files, e.g. 

C:\Program Files (x86)\EPiServer\CMS\7.0.586.1\Application\Util\styles\quicknavigator.css

Note that changes to this file will affect all sites on your machine/server using that EPiServer version.

The QuickNav is in the way; I'd like to move it a little... :)

Now that the UI comes as a NuGet package, it wouldn't be a good idea to change the quicknavigator.css file in the package, as this will be overwritten the next time I (or another developer) updates the packages.

Is there any good way to override the styles set in the package?

Why not just have your own .css files, the questions is if you want to go all the way to put it in your own module.

We've got a functionality on a site where it's a little outside the viewport untill your hover it using the css selector body #epi-quickNavigator