  Entanglement entropy satisfies a first law-like relation, which equates the
first order perturbation of the entanglement entropy for the region $A$ to the
first order perturbation of the expectation value of the modular Hamiltonian,
$\delta S_{A}=\delta \langle K_A \rangle$. We propose that this relation has a
finer version which states that, the first order perturbation of the
entanglement contour equals to the first order perturbation of the contour of
the modular Hamiltonian, i.e. $\delta s_{A}(\textbf{x})=\delta \langle
k_{A}(\textbf{x})\rangle$. Here the contour functions $s_{A}(\textbf{x})$ and
$k_{A}(\textbf{x})$ capture the contribution from the degrees of freedom at
$\textbf{x}$ to $S_{A}$ and $K_A$ respectively. In some simple cases
$k_{A}(\textbf{x})$ is determined by the stress tensor. We also evaluate the
quantum correction to the entanglement contour using the fine structure of the
entanglement wedge and the additive linear combination (ALC) proposal for
partial entanglement entropy (PEE) respectively. The fine structure picture
shows that, the quantum correction to the boundary PEE can be identified as a
bulk PEE of certain bulk region. While the \textit{ALC proposal} shows that the
quantum correction to the boundary PEE comes from the linear combination of
bulk entanglement entropy. We focus on holographic theories with local modular
Hamiltonian and configurations of quantum field theories where the \textit{ALC
proposal} applies.
