Although I can't say I've done this mid-stream for a URL, I'm doing something similar with YouTube channels, and I think it would be a similar analogy.
I originally set up a YouTube account with my last name as a user name. Then YouTube came out with channels, automatically making your user name your channel name, and my business name made more sense. So, new account new channel with the business name.
My various videos have always had links in the description fields to my other videos (as well as links to my web site). Since I now have the same videos on two channels, the links in the descriptions of the videos on the old channel point all to videos on the new channel (and of course the videos on the channel only point to new channel videos). Eventually, when the videos on the new channel have logged more views and "likes", I'll just remove them from the old channel and use that for personal stuff.
Sorry for the long story, but I think you could do something similar with your web sites. Make them identical and keep both running in parallel. But make the menu of your old site point (with absolute links) to those pages new site. It will be seamless to the visitor. That way you can milk the last few drops of Google juice from the old site. Who knows, maybe having your old site pointing to the new site will give you new site a bit of a head start on ranking.