it sounds like you might be in the right place
in the Hair Mat/Length channel Texture slot choose Effects/Projector (which is what I think you were talking about so, so far so good)
Click on Projector (now that it’s there) and it opens up and presents a new Texture slot where you want to put your image file – the settings below are the ones you would use to offset the image (in effect rotating around the sphere) with lots of additional controls to tweak the position/rotation/size etc.