I think there is a word missing here like "clerk" or "assistant". When you go to the hospital there is always an "admissions" process/ procedure to register the patient/ the personal data/ medical details on the hospital register. This can be done by a nurse or sometimes a member of the "admissions" team that are often found at reception desk of the hospital department. The telephone no. will be needed to give the patient the results of the blood test, later.

Yes, the reference is to an assistant who handles appointments, hospital admissions, etc. I also think the statement is grammatically incorrect and should have been: "Either I or my admitting will follow up . . . " I was taught both subjects should be treated grammatically as if there is only one. As we would have said, "I will follow up with you," not "Myself will follow up," the original reference to the speaker was incorrect. Although this next point isn't critical, also, typically, in English, we refer to ourselves last. So the statement should have even more correctly read: " Either my admitting or I will follow up. . . " Hope this isn't confusing and helps.

I think she meant of her co-workers that she admitted to this job will chat up with the patient later