There are three general options I can think of.
1) Early Stevens/Savage. At one time stevens and later savage made a number of small-frame over/under combo guns. These were first available in .22/.410 (I own one) and later a variety of calibers and gauges (.22/20, .22 WMR/20, etc). Savage named these the Model 24, with various models (ex: 24C). These were not a bad compromise. They worked relatively well, and I love em for squirrels (Use rifle when you can, shotgun for tree tops in situations whe you are concerned about were the bullet comes down).
2) later Savages in .223, 30-30, etc over 20 and 12. These later guns were chambered for substantially larger rounds. Teh guns themselves are also quite large, relatively cumbersome and kinda clunky. They are not cheap, and wieght is a big issue. Personal option: You can do better by buying a simple H&R in 30-30, .223, etc and then adding a shotgun barrel for $90, or buying a package gun with two barrels outright.
3) Various imports. Valmet once made a series of these in centerfire over 12. There are a few other isolated similar models. In general you either end up spending $1500++ or end up spending $400 for a brazilian/russian cheapo.
EDIT: 4) Springfield folding 22/410 or .22 hornet/410. Scarce and $$. Neat if you can find one. Dsicontinued/
The early stevens savage small frame rimfire over 410 or 20 are valid and useful. The others are too damned cumbersome and heavy to be worthwhile.
Fro.