  We consider the Heisenberg XXZ spin-$J$ chain ($J\in\mathbb{N}/2$) with
anisotropy parameter $\Delta$. Assuming that $\Delta>2J$, and introducing
threshold energies $E_{K}:=K\left(1-\frac{2J}{\Delta}\right)$, we show that the
bipartite entanglement entropy (EE) of states belonging to any spectral
subspace with energy less than $E_{K+1}$ satisfy a logarithmically corrected
area law with prefactor $(2\lfloor K/J\rfloor-2)$.
  This generalizes previous results by Beaud and Warzel as well as
Abdul-Rahman, Stolz and one of the authors, who covered the spin-$1/2$ case.
