How do I recover missing old photos from my iPhone?

Check the Hidden album.

Hide photos on your iPhone, iPad, Mac, or Apple Vision Pro with the Hidden album - Apple Support

If they are not in the main Library, in the Hidden album, or in the Recently Deleted album, that would suggest that they were permanently deleted.

Some pictures are shown by Photos but not managed by Photos. If you don't use iCloud Photos, or if you have pictures not managed by Photos, but you have Settings>Apple ID>iCloud Backup turned on, then you may be able to recover some data from the last 30 days or so at iCloud.com, but you would need to look at the date of the backup to be sure it's from before you lost the images. You should look here:

         View and manage iCloud device backups - Apple Support

This requires you to erase everything on your phone and replace it with the backup. I'm guessing that this sort of thing is what prompts travelingwoman333's warnings.

Check the hidden and recently deleted sections. Just a heads up, make sure to back up the data on your phone often and have trusted contacts as your Apple Recovery people. This past November, I accidentally deleted some important pictures and videos from the "Recently deleted" section, and Apple told me to erase the data on my phone. I did that and as a result, I lost access to my phone for 4 days.

Lesson learned: Write down the passwords to your Apple and other email accounts, have trusted people as your account recovery people, and watch out for what you delete. Slow down and think!