Genus one Birkhoff sections for geodesic flows on orbifolds

For $\mathcal{O}$ a hyperbolic orientable 2-orbifold of genus $g$ with at most $2g+6$ conic points, we prove that the geodesic flow on the unitary tangent bundle$\mathrm{T}^1\mathcal{O}$ admits a Birkhoff section whose genus is one. Together with a result of Minakawa, this implies that this flow is almost equivalent to the suspension flow of the $(\begin{smallmatrix}2\&1\\1\&1\end{smallmatrix})$-map on the torus.