This paper studies the second-order asymptotics of the discrete memoryless multiple-access channel with degraded message sets. For a fixed average error probability $ε\in(0,1)$ and an arbitrary point on the boundary of the capacity region, we characterize the speed of convergence of rate pairs that converge to that point for codes that have asymptotic error probability no larger than $ε$, thus complementing an analogous result given previously for the Gaussian setting.