Good idea, but I'd be so afraid of catching the cable as I push the new housing back in, even with a new cable. Make sure the cut's clean. I'd also consider using a toothpick and electrical tape to join 2 ends of the housing together and pull the new one through. Leave the new one long, and cut it once it gets to the far end.
Of course, the easiest is use full length external routing. Not as "pretty", but you don't ride to look pretty, do you?