I always use UPS - but I can't say that I use them enough to say whether they're any better than anyone else.
That said, I would never rely on a courier to treat a horn with as much care as is required - so I make sure the horn is very well packaged.
You don't really need to cork down the action - after all, it's designed to move at quite high speeds and to tolerate quite a bit of force. But you do need to ensure that the horn cannot move in the case. It's this movement that does the damage. If the box gets dropped, the impact is like hitting the horn with the inside of the case.
Use foam, rolled up pieces of bubblewrap, polystyrene blocks, packing peaunts...anything to softly wedge the horn in the case so that it doesn't move when you pick the case up and give it a shake.
It's often worth nipping down to your local tip to see whether anyone's dumped a sofa. The seat squabs will be a solid lump of 5 inch thick foam. Look out for polystyrene 'corners' (from large white goods packing boxes). These will probably need cutting to shape.
Use a box that's a bit larger than the case, at least a couple of inches all round - and either fill the space with soft packing material, or (preferably) mount the case on blocks of foam/polystyrene at each of its corners. Glue/tape the blocks in place - if the horn takes a tumble it may well survive the fall, but the blocks may be displaced...and then you're in trouble. Larger horns may need additional blocks to support the middle of the case. Use a water-based glue, as a solvent-based glue may melt foam/polystyrene.
This method places the case (and the horn) 'in suspension' and provides the most inefficient path for any impact force to travel through.
Bubble wrap is great stuff...but only when used in vast quantities. The horn needs to be wrapped so well that you could quite happily give the box a hefty kick and not worry that it will damage the horn.
So...simply wrapping the case in a box is no good. Wrapping two or three layers of bubblewrap around the case before boxing it up is no good either. And neither is using materials that will compress but won't recover (rolled up cardboard).
As for insurance, you really have to check the small print as many couriers will quite happily accept your money but fail to tell you that musical instruments aren't covered. Take lots of photos before you pack the horn, making sure that you have good shots of the area around the low Eb/C key guards, the front lip of the bell, the bell key table and the group of pillars at the top of the horn that hold the G/top E/top F# keys on. Take photos of the package too.
And then keep your fingers crossed.
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