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There is an iOS app called SimpleScreen which might work for what you want to do.

 
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My lock screen is all black and I put my email address at the bottom with an "if found, contact..... Reward"

Just get a photo editing app that will let you type text on an image . Save it and use as lock screen wallpaper.

Or you could set up a medical ID with info on it. Not necessarily medical info. That can be read by anyone who gets the phone.
 
An emergency or return-if-found number (different from the phone, obviously)? We had a phone that we wanted this info on - did it with an edited wallpaper.
Apple said:

Mark your device as lost​

  1. Open the Find My app and choose the Devices tab.
  2. Select your missing device.
  3. Scroll down to Mark As Lost and select Activate.
  4. Follow the onscreen steps if you want your phone number to display on your missing device, or if you want to enter a custom message asking the finder of your missing device to contact you.
  5. Select Activate.
Of course, may not be applicable/feasible for the OP's intent plus this requires the device has an Internet connection and Find My enabled.

emphasis mine
 
Should be an build in option to be honest. Android has it.
I did it manually in photoshop but it’s kind of a pain to get it in the proper position since the Lockscreen may Zoom the image, depending on your settings
 

Of course, may not be applicable/feasible for the OP's intent plus this requires the device has an Internet connection and Find My enabled.

emphasis mine

I'm aware of that - we just had different reasons for wanting the info available/visible on the phone.
 
An emergency or return-if-found number (different from the phone, obviously)? We had a phone that we wanted this info on - did it with an edited wallpaper.
No idea why it would need to be different from the phone... call the number, phone rings, so what? Leave a message. Voice messages are retrievable from anywhere.

When I travel, I put up my regular wallpaper, but with my name, my phone number, and my email address overlaid.
 
No idea why it would need to be different from the phone... call the number, phone rings, so what? Leave a message. Voice messages are retrievable from anywhere.

When I travel, I put up my regular wallpaper, but with my name, my phone number, and my email address overlaid.
I hear you, but like I said, we had good reasons that I won't go into here.
 
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