Mate, it’s a ball-ache that’s for sure….However there are a few techniques. You say the outer is not through the frame, in which case it’s either in a channel especially for the cable, or more than likely just going solo through the frame and a complete pain in the A! If it’s the latter, get a magnet and as you feed the cable in, put the magnet on the frame so it connects with the cable through it (should here the cable moving against the frame as you you guide it down). After that you’ll need the guile of a fox to guide it down to the exit point which I hope is a decent size to get it out either luckily in that it just comes out, or with long nose pliers if you can fit them in.
If the housing is all the way through, then leave the inner in situ and pull out the outer completely (leaving only the inner cable in place). After that you’re not bothered about kinking the inner so feed the new outer back through until in place. Then remove the inner cable and replace it and vwal-the-damn-la! 😀 …Believe me you’ll be swearing whichever option you choose, but hope it helps.