So you need other identifiers or differentiators to get a better match.
Some organizations have business rules about matching that only allows them to expose patient data to other providers if they get one and only one match. Thye also have business rules about not displaying any data for patients whose charts are restricted outside their practice.
Combine these two issues and you have a tricky challenge that is easy to get wrong.
How do you implement the patient identity search? Do you search for only patients with unrestricted charts, or do you search with both but only display if you get unrestricted results? You have to search with restricted patients included! Consider: If an identity collision occurs, regardless of whether it occurs for a patient with a restricted chart or not, you still have to detect it! If you were to just search the unrestricted results, and there were two John Smith's whose identity collided, when someone tried to access data for the one with the restricted chart they would get data for the wrong John Smith 100% of the time.
So, you cannot restrict the identity search, but you have to restrict what it reports.
This will impact patients whose identity happens to collide with those that have restricted access to their chart. That is where other identifiers or data (such as Mother's maiden name, email or phone number) can help differentiate patients.