  We explore a number of problems related to the quadratic Chabauty method for
determining integral points on hyperbolic curves. We remove the assumption of
semistability in the description of the quadratic Chabauty sets
$\mathcal{X}(\mathbb{Z}_p)_2$ containing the integral points
$\mathcal{X}(\mathbb{Z})$ of an elliptic curve of rank at most $1$. Motivated
by a conjecture of Kim, we then investigate theoretically and computationally
the set-theoretic difference $\mathcal{X}(\mathbb{Z}_p)_2\setminus
\mathcal{X}(\mathbb{Z})$. We also consider some algorithmic questions arising
from Balakrishnan--Dogra's explicit quadratic Chabauty for the rational points
of a genus-two bielliptic curve. As an example, we provide a new solution to a
problem of Diophantus which was first solved by Wetherell. Computationally, the
main difference from the previous approach to quadratic Chabauty is the use of
the $p$-adic sigma function in place of a double Coleman integral.
