  To which degree the average entanglement entropy of midspectrum eigenstates
of quantum-chaotic interacting Hamiltonians agrees with that of random pure
states is a question that has attracted considerable attention in the recent
years. While there is substantial evidence that the leading (volume-law) terms
are identical, which and how subleading terms differ between them is less
clear. Here we carry out state-of-the-art full exact diagonalization
calculations of clean spin-1/2 XYZ and XXZ chains with integrability breaking
terms to address this question in the absence and presence of $U(1)$ symmetry,
respectively. We first introduce the notion of maximally chaotic regime, for
the chain sizes amenable to full exact diagonalization calculations, as the
regime in Hamiltonian parameters in which the level spacing ratio, the
distribution of eigenstate coefficients, and the entanglement entropy are
closest to the random matrix theory predictions. In this regime, we carry out a
finite-size scaling analysis of the subleading terms of the average
entanglement entropy of midspectrum eigenstates when different fractions $\nu$
of the spectrum are included in the average. We find indications that, for
$\nu\rightarrow0$, the magnitude of the negative $O(1)$ correction is only
slightly greater than the one predicted for random pure states. For finite
$\nu$, following a phenomenological approach, we derive a simple expression
that describes the numerically observed $\nu$ dependence of the $O(1)$
deviation from the prediction for random pure states.
