to work for a clinic,hospital,firm etc.
.......... at a clinic,hospital,firm
............in a clinic,hospital,firm
for a clinic You are being paid by the clinic. You may actually work somewhere other than the clinic, like a university.
at a clinic You are employed in the immediate vicinity of or inside the clinic. You may work for the clinic as a doctor, or you may sell hot dogs in the lobby.
in a clinic Similar to at a clinic except you are always inside the clinic.
There is also by a clinic which implies that you work somewhere other than the clinic itself.