Yes, your old iPhone 8 or iPhone SE (2020) case will fit the iPhone SE (2022) as they all use housing with absolutely the same dimensions and layout. Their cases are interchangeable.
Some noticeable improvements
The iPhone SE 3 (2022) has some fabulous new features. The new handset features a powerful A15 Bionic chip. It has an upgraded camera, with Smart HDR 4, Deep Fusion, Photographic Styles, and more. You'll be able to use it longer between charges due to improved battery life. The iPhone SE 3 can access a faster 5G connection so you can do more with less lag time, though it's important to note that it only supports Sub-6 5G and not the faster mmwave like the iPhone 13 family. The third-generation iPhone SE has a lot to offer new iPhone users, people on a budget, or folks who want a smartphone that "just works" and don't want a bunch of bells and whistles.
New phone, same form
Apple has kept the same compact form factor and dimensions across several iPhone models, including the Home Button and Touch ID. The iPhone SE 3 measures 5.45-by-2.65-by-0.29 inches, or 138.4-by-67.3-by-7.3 millimeters, just like the iPhone SE 2 (2020). The buttons and ports remain in the same locations as well. The camera area is also the same, so that won't be an issue either. Incidentally, the iPhone 8 and iPhone 7 also share the same dimensions, so if you're coming from one of those older iPhones, you can still use the same case. Of course, if you do want a new case anyway, it may be a good idea for you to try out our Custom Name Wood Case for iPhone SE 3 (2022). Click the photo.