Your effort to contact the priest has to be in line with the urgency of the situation.
The hospital, or a parishioner even, could call using the reserved line when there was a life and death emergency. I responded to those day or night. Frankly, there were times when people grossly abused that particular telephone line – and that was really not appreciated. On the other hand, when a parishioner was undergoing an operation in the near future and approached me after Mass to arrange for the sacrament of the sick, I would always suggest just doing it then and there.
I join the others who have recommended that you telephone the rectory and ask for an appointment when a priest can hear your confession and anoint you.
Alternatively, if the procedures are being done at a Catholic hospital, you could reach out to the pastoral care department and make an arrangement to be seen there.
Personally, I will tell you that I tended always to err on the side of generosity regarding the prudential judgment a priest exercises to anoint. I had several sad experiences in my first years as a priest involving patients having complications with relatively routine procedures and, frankly, it affirmed me in my tendency to be broad in understanding “danger,” which is the criteria for the sacrament of the sick in canon law; this is one of those moments when I read the canon both for the language it uses and, more specifically, the language it does not use. To this day I will use my pastoral discretion to anoint when other priests might choose not to do so…and may even not agree with my decision. Fortunately, while it is their prerogative not to agree with my decision, it nevertheless remains my prerogative to proceed.
The fact that you are undergoing examinations would not, in itself, necessarily indicate anointing or that this is a pastoral emergency.
