"Need For Urgent Appointment At A Hospital Clinic"

I've been attending a hospital clinic for some time and have recently changed to a different medication. I have a return appt but that's not until the end of the year. Apart from a feeling this medication isn't working, I also have a bloodshot eye so I phoned up to see if I could attend the clinic urgently and was initially offered an appt several weeks away. I said that wasn't early enough so they looked for cancellations and came back with one for the following day .  

Now that's worked out OK but what if there had been no cancellation? There are going to be situations where patients need to attend clinics between appointments but there seems to be no mechanism to accommodate patients in such circumstances and receptionists don't appreciate the need. 

So if you need to see a clinician what are you supposed to do?