  We complete the computation of all $\mathbb{Q}$-rational points on all the
$64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is
hyperelliptic. To achieve this, we use a combination of various methods, namely
the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty,
and the bielliptic quadratic Chabauty method combined with the Mordell-Weil
sieve. Additionally, for square-free levels $N$, we classify all
$\mathbb{Q}$-rational points as cusps, CM points (including their CM field and
$j$-invariants) and exceptional ones. We further indicate how to use this to
compute the $\mathbb{Q}$-rational points on all of their modular coverings.
