What you hope for is not in general possible.
For example, the overtwisted contact structure $\xi_{-\frac{1}{2}}$ on $S^3$ with $d_3 = -\frac{1}{2}$ (ie. the overtwisted contact structure in the same homotopy class as $\xi_{\rm{std}}$) has a supporting open book which is a thrice-punctured sphere, by [Etnyre–Ozbagci, Lemma 5.5]. If we could replace this open book with one that has a once-punctured torus page, the binding would be a genus-1 fibred knot in $S^3$. Thus, the binding is either the right-handed trefoil, the left-handed trefoil, or the figure-eight knot, which support $\xi_{\rm{std}}$, $\xi_{\frac{3}{2}}$, and $\xi_{\frac{1}{2}}$, respectively.