As a specialist's office, this is the method we use. Over the course of about a week, every time I make the attempt, I document the response/lack thereof, and send it back to myself. I try the patient's preferred number first and leave a message, wait a day or two, then try their preferred and secondary number the next time and leave a message on both. Wait another day or two and if that doesn't work, I call their emergency contact to make sure the patient is OK and see if they can have the patient call me. If the patient doesn't call me back based on the conversation with the EC, I document this as well. We also use texting from Updox, since not all patient's will answer their phone and it's not hard to answer a text at work; even if it's just to let me know they got my message(s). TBH, some people don't have the decency to even do that much. When I do send a text that I'm trying to reach them, I document that in the notes I've forwarded to myself by copy/pasting it from Updox; both the body of the message and the date/time. Every time I send it to myself, it has a new date/time stamp. At the end, I have a well documented list of attempts to contact the patient that I save to the chart and they can't say no one tried to reach them. At this point, I then notify the PCP and/or referring office in writing and cc them on whatever report I'm trying to reach the patient about.
I also use this method when trying to reschedule appointments cancelled with the Amazing Reminders system.
Hope this helps.