Thanks, Quick, for recommending the Morgan adjustable pad.
Here it is with a 10 minute installation job on a Mossberg shotgun.

I have an inch more in drop to the stock. With the mouse pad/ammo carrier combination I have a really comfortable cheekweld. It may look goofy, but it has made the shotgun a lot more comfortable.
If anyone thinks that the Handi, or another type, doesn't have enough drop, then I encourage him to get one of these.
I am going to buy a second one for the 357 Handi.