Detect paid user after mobile app reinstall

if a user pays for your app and then reinstalls, perhaps they got a new phone. Is it possible to detect the user outside of linking the payment history to email/phone number?

Basically I plan on having only 1 app (free with in app purchase) instead of 2 apps (1 free & 1 paid). If a user downloads the free app does inapp purchase and later reinstalls I wish for them to be able to have access to their purchase.

Any one know how?