  Given $k\ge3$ and $1\leq \ell< k$, an $(\ell,k)$-cycle is one in which
consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We
study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which
do not contain hamiltonian $(\ell,k)$-cycles, but once a new edge is added,
such a cycle is promptly created. It has been conjectured that this number is
of order $n^\ell$ and confirmed for $\ell\in\{1,k/2,k-1\}$, as well as for the
upper range $0.8k\leq \ell\leq k-1$. Here we extend the validity of this
conjecture to the lower-middle range $(k-1)/3\le\ell<(k-1)/2$.
