Note that Paul generally lacks any coherent “new covenant theology,” since he only speaks of the “new covenant” here (2 Cor 3:6) and in 1 Cor. 11:25 in reference to the Lord’s Supper. 

His lack of emphasis on the “new covenant” would seem to indicate that the fulfilment of Jer. 31:31–34 did not substantially alter his overarching view of redemptive history. Rather, the death of the Messiah was simply understood within his preconceived eschatological framework, akin to fulfilment of the death of the Messiah in Isaiah 53 (which did not mean chapters 40–52, or 54–66, were being spiritually realized). Such selective fulfillment of verses (leaving intact surrounding eschatological expectations) is seen throughout the NT in relation to passages such as Psalm 110, Zechariah 9, Joel 2, Isaiah 61, etc.  - John P. Harrigan