  We consider edge decompositions of $K_v^{(3)}-I$, the complete 3-uniform
hypergraph of order $v$ minus a 1-factor (parallel class, packing of $v/3$
disjoint edges). We prove that a decomposition into tight 6-cycles exists if
and only if $v\equiv 0,3,6$ (mod 12) and $v\geq 6$; and a decomposition into
tight 9-cycles exists for all $v\geq 9$ divisible by 3. These results are
complementary to the theorems of Akin et al. [Discrete Math. 345 (2022)] and
Bunge et al. [Australas. J. Combin. 80 (2021)].
