Thank you for contacting us.
The only method available is using the microSD Card to extract the image (backup), or to restore it on another Intel Edison. So, the best suggestions that I can give you are:
1. The SD card pins are available in the Mini Breakout board, so you can try to implement an SD card interface. (Check this thread: https://communities.intel.com/thread/68367)
2. You can use an Arduino Expansion board that already comes with the SD card slot (Easy way).
3. You can use another SD shield with your Edison module, like this one: https://www.sparkfun.com/products/13041
Once you have the SD card interface you can follow this thread to accomplish your goal: https://communities.intel.com/thread/55807
I hope you find this information helpful.
Have a nice day.