We study the second-order asymptotics of the Gaussian MAC with degraded message sets. For a fixed average error probability epsilon in (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 epsilon. We do so by elucidating clear relationships between the rate pairs achievable at large blocklengths and the local second-order asymptotics, i.e. the second-order behavior of these rate pairs in the neighborhood of a boundary point. 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.