Anointing of the sick may take place during Mass, as most contemporary Catholics know and experience it. What does the rite say? If the sick person and her or his family and caregivers are to receive Communion, celebrating the Mass is a logical option, pastoral circumstances permitting.
131. When the condition of the sick person permits, and especially when communion is to be received, the sacrament of anointing may be celebrated within Mass. The following rite provides for such a celebration, which takes place in a church or, with the consent of the Ordinary, in a suitable place in the home of the sick person or in the hospital.
The bishop is responsible for any anointing that takes place outside of a church. Anointing within Mass covers PCS 135-148.
132. This rite may be used to anoint a number of people within the same celebration (see PCS 108-110). It is especially appropriate for large gatherings of a diocese, parish, or society for the sick, or for pilgrimages. Since the liturgical texts appear in the plural, they must be adapted in gender and number for a celebration in which one person is anointed.
Anointing at Mass most often takes place in the parish setting. How many dioceses provide for such celebrations? Outside of the cathedral parish, celebrations as a diocese are unknown to me. Do any of your dioceses have them?
133. The priest should ensure that the sick who wish to celebrate the sacrament of penance have a convenient opportunity to do so before Mass.