After supper may suggest that the main meal separated the breaking of the bread from the drinking of the cup (see Luke 22:20).
The new covenant is God's promise to forgive sins because of Jesus' sacrificial death (compare Jeremiah 31:31–34; 32:40; Luke 22:20; Hebrews 7:22; 8:8–10; 9:15; 10:12–18; 12:24; 13:20). The blood of a sacrifice confirmed an agreement or covenant (see Exodus 24:8; Zechariah 9:11; Hebrews 9:12; 13:20; compare Romans 3:25; 5:9).