I haven't discovered how to do this yet either. I had an iPhone 5 that I was manually entering data in for the last week (flights, weight, etc.), and I'd love to transfer that stuff over to my new iPhone 6 Plus, but I can't find any software options to do that, and it didn't happen with my iCloud backup / restore from the old phone to the new one.
It's only a few days' worth of data so I could just re-enter it manually on the new phone, but I'd love to find a shortcut before going that route...