The strong coupling expansion for staggered fermions allows for Monte Carlo simulations using a dual representation. It has a mild sign problem for low values of the inverse gauge coupling
$\beta$, hence the phase diagram in the full $\mu_B - T$ plane can be evaluated. We have extended this framework to include
$\mathcal{O}(\beta)$ and $\mathcal{O}(\beta^2)$ corrections, by mapping the degrees of freedom to a vertex model. We present results on the $\beta$-dependence of the chiral critical point from those simulations.