Converse and achievable results on the third-order coding rate for universal fixed-to-variable source coding are presented. The authors previously introduced a new universal non-prefix Type Size code for memoryless sources in which codewords lengths are chosen in ascending order of type class sizes. A new converse proves that no universal code can do better than the Type Size code in terms of third-order coding rate. Moreover, new results are obtained for prefix codes, showing that the prefix constraint strictly increases achievable third-order coding rates for universal codes.