There is a trick where you don’t set your L limit until after you attach the cable. I have not needed to do this, but it is used on new front derailleurs where there can be no slack.
Turn the L limit a couple(?) of turns, keep your barrel adjuster in the middle, get the cable tight as you can and then back out the L limit screw.
Others here can give you better detail. I’m not sure how much you need to move the cage.
John