Packing well is a must and distance hardly ever matters because most of the times a package go through a similar amount of changing of hands while only the length of the haul in between the main shipping points changes.
Regardless whether you send a horn to the next county or across the world one should always pack well.
My wife and I have discovered that one of the best isolation material to put all around a case are egg cartons. Which we save especially for this purpose.
The horn inside a case needs to be padded in the case so that it cannot shake, rattle and roll
anyway. Then you put your case in a large carton box (or you make one around the case using smaller carton boxes) making sure that is completely suspended on all sides with egg cartons all around the inner case.
The saxophone this way will cushion any hit (but the saxophone inside the case has to be absolutely incapable of moving in the case!!)
Egg cartons are lightweight but (when closed) very sturdy (they are made to protect eggs after all).
We have been complimented many times for our shipping technique.
Yes, as you can see from the calculator that I have provided.