The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of Hàn, Person and Schacht who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an $n$-vertex $3$-graph is $\left(\frac{5}{9}+o(1)\right)\binom{n}{2}$. In this paper we improve on this result, giving a family of degree sequence results, all of which imply the result of Hàn, Person and Schacht, and additionally allow one third of the vertices to have degree $\frac{1}{9}\binom{n}{2}$ below this threshold. Furthermore, we show that this result is, in some sense, tight.