Quadratic torsion orders on Jacobian varieties

We establish the existence of hyperelliptic curves of genus $g\ge 2$ defined over $\mathbb{Q}$ whose Jacobians possess rational torsion points of order $N$ where $N=4g^2+2g-2$ or $4g^2+2g-4$. For $N=2g^2+7g+1$, we introduce a $1$-parameter family of hyperelliptic curves of genus $g$ over $\mathbb{Q}$ with a rational torsion point of order $N$ on their Jacobians.