I shipped an expensive (to me anyway) Taylor from Cleveland to New Jersey about 10 years ago. Did some searches and found some videos explaining how to do this very thing. Could be wrong but I think I used a method shown by Erlewine.
From what I can remember, I used a little bit of bubble wrap to fill some space by the headstock, loosened the strings about a step down, and then closed up the case. From there I slid the case into the original Taylor box that I still had and filled all the open space with crumpled newspaper. Sealed that up and then wrapped that box with bubble wrap and slid it into another box.
I still have the pictures I took to document the packing in case something went wrong somewhere but that is how I did it to the best of my recollection.