First, look at a process a patient is going through and map it out. Below is an example of a basic flow map .
Use different symbols to represent different types of process. In this case the ovals are for administrative steps and the rectangles are for clinical. The arrows in between are for waiting. Since the map can be drilled down to an insane amount of detail it's better to first get a global overview and look for the bottle neck. The bottle neck will have either a high average time in process or a high degree of variation with a long wait before it. Once you find the bottle neck work to map it out as well.
It is not necessary to follow individual people through the clinic 20 or 30 times to make the measurements. Instead, get however is doing the process (better yet an objective third party) to write down the times. Or sit at one process and measure people going through it repeatedly. Eventually you will arrive at an average time in the process and you'll have a range. For the more mathematically inclined put the figures into a spreadsheet and calculate the standard deviation.
By creating process flow maps, finding the bottlenecks and working to resolve them you'll find that the number of people that can go through the system improves as does their satisfaction. The process is not about spending less time with people it is about making that time flow better.