When you move to a new phone it may not have your authenticator app installed immediately.
When the verification code is next required to log in, you will need to have the authenticator app accessible. This should be downloaded to your new device from your usual app store.
You will need your old and new phone for this to work.
There will be a process in place to transfer the account to a new device, the example given below is for Google Authenticator, but others should work similarly.
On your Old phone, please open the authenticator app, select the options button (likely to be 3 horizontal lines or dots). This should allow an option to Export accounts.
The export button will bring up a QR code on screen.
On your New phone, select the option to add an account to your authenticator app. This will give the chance to either enter the details manually or scan a QR code, please select the scan option.
Scan the code on your old phone and the account should immediately appear on the new device.
Other Authenticator apps may have different processes. For example, Microsoft Authenticator requires you to go to the Settings area (also 3 vertical dots usually), then adjust the Back up settings. Once cloud backup is enabled, the data should be accessible from the app on your new phone.