The Trinity in relief

There's a simple reason why the Trinity lies in the background of the OT but the foreground of the NT. That's because salvation is Trinitarian. The Atonement is Trinitarian. It unites the work of the Father, Son, and Spirit. The Son doesn't die for people at random. Rather, he consented to become Incarnate, become the Savior, on condition that his redemptive death would secure the salvation of all those the Father gave him (Jn 6,10,17). Likewise, the Father adopts and justifies the redeemed. So the Father and Son work in tandem. Likewise, the Spirit renews and preserves all those whom the Son redeems. So the Son and Spirit work in tandem. 

It was therefore inevitable that the Trinity would come into relief when the Atonement came into relief. The OT was preparatory, but when the Atonement is actually implemented, the three players will come to the fore.