Yes, you remove the spool to reload it.
This question arises because when you screw the spool back on the trimmer head, it "unwinds" the string that was wound in the direction of the arrow- that seems wrong. You can't completely reinstall the spool without having a bunch of loose string.
The new string supply container is made to let string feed out as it is pulled - so it's easy to restring by turning the spool rather than winding the string after it's pulled out of the container.
Guess I'll try it both ways and see which works best.