Find another iPhone I own but isn’t on service plan

Your service plan has nothing to do with Find My iPhone.

You can log in to https://icloud.com with the Apple ID that was used on the old iPhone for iCloud and it should show up in the Find My App there if its turned on and can connect to the internet.

This other phone does not need to be on an AppleCare+ service plan, but it would need to be logged into with the same Apple ID which would associate it with that account and listed as one of your device on your Apple ID Device List.

A quick note to point out that If the battery has run down on the older iPhone, it is not going to appear on Find My....even if you are doing everything else correctly.