I can get to the bottom and top of my refrigerator since it has a vent cover at the bottom and top of the outside wall. With this access, I could fish in a new water line between the solenoid valve and the icemaker feed by connecting the new line to the old line at the bottom and pulling the old line out from the top.
I noted at the hardware store that the soft, clear plastic tubing used for these icemaker feeds carried a tag -
WARNING: Do not use for icemakers! That's one of those things that make you go hmmmmm....
So, I used the hard, translucent line that's used on home icemakers to replumb mine. It works fine.
Rusty