  For $0\leq \ell <k$, a Hamiltonian $\ell$-cycle in a $k$-uniform hypergraph
$H$ is a cyclic ordering of the vertices of $H$ in which the edges are segments
of length $k$ and every two consecutive edges overlap in exactly $\ell$
vertices. We show that for all $0\le \ell<k-1$, every $k$-graph with minimum
co-degree $\delta n$ with $\delta>1/2$ has (asymptotically and up to a
subexponential factor) at least as many Hamiltonian $\ell$-cycles as in a
typical random $k$-graph with edge-probability $\delta$. This significantly
improves a recent result of Glock, Gould, Joos, K\"uhn, and Osthus, and
verifies a conjecture of Ferber, Krivelevich and Sudakov for all values $0\leq
\ell<k-1$.
