  We consider the generalised Jacobian $J_{0}(N)_{\mathbf{m}}$ of the modular
curve $X_{0}(N)$ of level $N$, with respect to the modulus $\mathbf{m}$
consisting of all cusps on the modular curve. When $N$ is odd, we determine the
group structure of the rational torsion
$J_{0}(N)_{\mathbf{m}}(\mathbb{Q})_{tor}$ up to $2$-primary and $l$-primary
parts for any prime $l$ dividing $N$. Our results extend those of Wei--Yamazaki
for squarefree levels and Yamazaki--Yang for prime-power levels.
