I learned that many users who have upgraded their iPhones to iOS 16 have the same issue. As your iPhone detects the SD card reader, it means that the culprit is not the connection.
You can try completely restarting your iPhone to refresh iOS and check if it can detect your SD card reader now.
If it still fails, try the following way:
- Insert the SD card into the card reader.
- Plug the card reader into your iPhone.
- Tap on the Files app > iCloud Drive's "Browse" > Browse's "No Name".
Now, you can see the files on the SD card through "No Name" on your iPhone. When an iOS update is available, please update your iPhone as soon as possible.