  Answering a question of Zureick-Brown, we determine the cubic points on the
modular curves $X_0(N)$ for $N \in \{53,57,61,65,67,73\}$ as well as the
quartic points on $X_0(65)$. To do so, we develop a "partially relative"
symmetric Chabauty method. Our results generalise current symmetric Chabauty
theorems, and improve upon them by lowering the involved prime bound. For our
curves a number of novelties occur. We prove a "higher order" Chabauty theorem
to deal with these cases. Finally, to study the isolated quartic points on
$X_0(65)$, we rigorously compute the full rational Mordell--Weil group of its
Jacobian.
