Now, however, Jesus has received a much more excellent ministry, just as the covenant He mediates is better and is founded on better promises.
6. But now has he obtained a more excellent ministry, etc. As he had before inferred the excellency of the covenant from the dignity of the priesthood, so also now he maintains that Christ’s priesthood is more excellent than that of Aaron, because he is the interpreter and Mediator of a better covenant.
In this part of the chapter, the apostle illustrates and confirms the superior excellency of the priesthood of Christ above that of Aaron, from the excellency of that covenant, or that dispensation of the covenant of grace, of which Christ was the Mediator (Heb 8:6): his ministry is more excellent, by how much he is the Mediator of a better covenant.
Commenting on Hebrews 8:6-13
But now hath he obtained a more excellent ministry,.... Christ has a ministry, he is the minister of the sanctuary, Heb 8:2 he has "obtained" this ministry of his Father; he was called unto it and engaged in it by him; and he has "now" obtained it; for though he was called to it from eternity, it was in time he came an high priest...