Many details for this answer would require knowing exactly the equipment you own and details about how the Airtag was relocated. Your iPhone is an older model with an old iOS version on it. It may not support NFC required for precise location*. It doesn't have NFC so it can't even tell if it is right next to you or just within Bluetooth range. Once the Airtag is outside of Bluetooth range (a few hundred feet at best) it either has to be able to connect to another phone using Bluetooth, or all you will get is its last known location which is the same as when it was next to you..
Airtag Overview --> https://www.apple.com/airtag/
Use the Find My app to locate a missing device or item
https://support.apple.com/HT210515
*iOS 14 - Features - Apple [ https://www.apple.com/ios/ios-14/features/#footnote-14 ]. The text below is in reference to NFC in iOS 14:
"Supported automatically on iPhone XS, iPhone XS Max, iPhone XR, iPhone 11, iPhone 11 Pro, and iPhone 11 Pro Max.
Supported after turning on NFC scan mode from Control Center on iPhone 7, iPhone 7 Plus, iPhone 8, iPhone 8 Plus, and iPhone X."