One trick we have learned over the years. Get your rifle Priority handling. Baggage insurance (above the normal coverage) can be purchased when you get your ticket. Just make sure you get it from the airline not a third party. This will give you priority handling as the carrier now has to treat it differently to minimize their potential liability. Select your flights and then buy the ticket by phone so you can tell the representative what you need. Don't try to do this at check in.
Another way is to fly Business/ First. You get Priority status with your ticket. Usually it is two free checked bags. The weight and size limit might also be more generous. This becomes more important for international flights with more onerous weight restrictions. Be aware that your rifle case may be sturdy, but that robustness can result in an overweight bag. About 25 years ago I invested in a welded, aluminum case from ICC - Impact Case Company. It has a long rod connecting four hasps and has never let me down. Also I firmly believe that you should not cut the foam out in the shape of the rifle. It is much more secure to have the foam form a crush fit around the rifle so that it can't slide around.
If two or three shooters are travelling together, only one person needs the upgrade and can carry the others' rifles. The third checked bag will cost cash, but still comes out cheaper. The others can carry your (non - rifle) checked luggage. An enhanced ticket also lets you have more and larger carry - ons. As always, make sure you have a change of clothes and all necessities in the cabin with you.
Just my experience. Learned the hard way.