This is just a shot in the dark reply, so it might be totally impractical. However, I couldn't help notice that the Camino de Levante (which starts in Valencia and hits the Plata) passes south of Madrid. So, with a bit of research, you might just be able to head south and hit the Levante. On the Plata, if you go via Ourense, you don't repeat any of the Frances. I checked out the Levante because I thought that it would be a pretty cool trip to start in Valencia and end up in Finnesterre.
Update: I found out more about the Levante and it passes through Toledo. I am not sure how one could go about getting from Madrid to Toledo on foot. However, Toledo might make an alternative starting point is you wanted to start near Madrid, but not repeat part of the Frances.