"at the clinic" is correct. Where are those people outpatients? They are outpatients at the clinic (across the street).
You can also say:
a) 'outpatients from the clinic.', indicating that they come to your house from the clinic.
b) 'outpatients of the clinic.', indiicating they are the clinic's outpatients. However, it is more natural to say 'rented the upstairs rooms to the clinic's outpatients.'.
'outpatients for the clinic.' is not correct. You have rooms for the clinic's patients to stay in, but they are not 'outpatients for the clinic.'
'the clinic' is not the same as Johns Hopkins Hospital; it is only a part of the whole hospital complex. (I have never been there but I am assuming that is correct)
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