  Bruin--Najman and Ozman--Siksek have recently determined the quadratic points
on all modular curves $X_0(N)$ of genus 2, 3, 4, and 5 whose Mordell--Weil
group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3,
4, and 5 and positive Mordell--Weil rank. The values of $N$ are 37, 43, 53, 61,
57, 65, 67 and 73. The main tool used is a relative symmetric Chabauty method,
in combination with the Mordell--Weil sieve. Often the quadratic points are not
finite, as the degree 2 map $X_0(N)\to X_0(N)^+$ can be a source of infinitely
many such points. In such cases, we describe this map and the rational points
on $X_0(N)^+$, and we specify the exceptional quadratic points on $X_0(N)$ not
coming from $X_0(N)^+$. In particular we determine the $j$-invariants of the
corresponding elliptic curves and whether they are $\mathbb{Q}$-curves or have
complex multiplication.
