  We consider the XXZ spin chain, characterized by an anisotropy parameter
$\Delta>1$, and normalized such that the ground state energy is $0$ and the
ground state given by the all spins up state. The energies $E_K =
K(1-1/\Delta)$, $K=1,2,\ldots$, can be interpreted as $K$-cluster break-up
thresholds for down spin configurations. We show that, for every $K$, the
bipartite entanglement of all states with energy below the $(K+1)$-cluster
break-up satisfies a logarithmically corrected (or enhanced) area law. This
generalizes a result by Beaud and Warzel, who considered energies in the
droplet spectrum (i.e., below the 2-cluster break-up).
  For general $K$, we find an upper logarithmic bound with pre-factor $2K-1$.
We show that this constant is optimal in the Ising limit $\Delta=\infty$. Beaud
and Warzel also showed that after introducing a random field and disorder
averaging the enhanced area law becomes a strict area law, again for states in
the droplet regime. For the Ising limit with random field, we show that this
result does not extend beyond the droplet regime. Instead, we find states with
energies arbitrarily close to the $(K+1)$-cluster break-up whose entanglement
satisfies a logarithmically growing lower bound with pre-factor $K-1$.
