I worked for doctors for years and they all have an answering service in case of urgent problems prior to or after normal business hours. The service can then either advise of the time the office will open, immediately contact the doctor or his/her on-call doctor or if an emergency connect the caller to 911.
I worked for a doctor who would assess the apparent mental stability of the patient at the first appointment (he is an oncologist) and if they seemed fine give them his cell number.
He actually had a patient call him at home at 3:00 am to ask what time her appointment was that day.
After that incident he became more selective about giving out his number to patients.