This paper studies the second-order asymptotics of the Gaussian multiple-access channel with degraded message sets. For a fixed average error probability epsilon is an element of (0, 1) and an arbitrary point on the boundary of the capacity region, we characterize the speed of convergence of rate pairs that converge to that boundary point for codes that have asymptotic error probability no larger than e. As a stepping stone to this local notion of the second-order asymptotics, we study a global notion, and establish relationships between the two. We provide a numerical example to illustrate how the angle of approach to a boundary point affects the second-order coding rate. This is the first conclusive characterization of the second-order asymptotics of a network information theory problem in which the capacity region is not a polygon.