  We give refined methods for proving finiteness of the Chabauty--Coleman--Kim
set $X(\mathbb{Q}_2 )_2 $, when $X$ is a hyperelliptic curve with a rational
Weierstrass point. The main developments are methods for computing Selmer
conditions at $2$ and $\infty$ for the mod 2 Bloch--Kato Selmer group
associated to the higher Chow group $\mathrm{CH}^2 (\mathrm{Jac}(X),1)$. As a
result we show that most genus 2 curves in the LMFDB of Mordell--Weil rank 2
with exactly one rational Weierstrass point satsify $\# X(\mathbb{Q}_2 )_2
<\infty $. We also obtain a field-theoretic description of second descent on
the Jacobian of a hyperelliptic curve (under some conditions).
