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I couldn't find these steps online before, so here's how to setup an Access Code on your new phone's HMRC app:
(These steps are written for the desktop gov.uk website as at June 2022. The site may later change and the mobile site may differ.)
1. Sign into your "Personal Tax Account": https://www.gov.uk/personal-tax-account
2. Go to "Your Profile": https://www.tax.service.gov.uk/personal-account/your-profile
3. Change your "Sign-in details" under Settings
4. Click on "How you get access codes"
5. Click on "Change" for the existing access code and scan the QR code from your HMRC app or other authenticator app
(or create a new access code with "Add another way of getting access codes" and then remove the old access code)