Are you comfortable editing text files? If so:
- open the gpx file and find the beginning of the list of points;
- cut/paste the points from before where you want the route to start at the end of the list of points (not at the end of the whole file);
- save.
The fiddly bit is identifying where within the file you want the new start point to be. Generally, you can get close by looking at how far through the route the new start point is, on a graphical editor (so if it's 30% of the way round, cut/paste about 30% of the points then save / load / check / re-edit as required. Sounds fiddly, but in practise it's a couple of minutes to do.
If there is a graphical editor which will do what you want, that's a better option, but I've not come across one.
Alternatively, put the existing route up on one side of your screen and just make a new route, using the old route purely as a visual reference. The risk there is that if you don't zoom in far enough it's possible that you won't choose exactly the same roads if it's a road-dense area. The text editor option is the easiest / quickest.