Suppose $f$ is a Lipschitz continuous real-valued function over a bounded domain $\Omega \subset \mathbb{R}^d$ with smooth boundary, and let $\overline{f} := \frac{1}{|\Omega|}\int_\Omega f(x) dx$. Is it possible to upper bound $\|f-\overline{f}\|_{L^\infty(\Omega)}$ in terms of $\|\nabla f\|_{L^2(\Omega)}$ using a generalized Poincare inequality or a related inequality? Of course, standard Poincare gives: $$\|f-\overline{f}\|_{L^2(\Omega)} \leq C\|\nabla f \|_{L^2(\Omega)}$$ but can this be "upgraded" to an $L^\infty$-norm bound in terms of $\|\nabla f\|_{L^2(\Omega)}$ using the fact that $f$ is Lipschitz?

This question asks something similar: Bounding supremum norm of Lipschitz function by L1 norm and I think it might be possible to string together an answer from the replies there. But I wonder if there is a more direct approach, using, for example, the Gagliardo-Nirenberg interpolation inequality. However, all versions of Gagliardo-Nirenberg for bounded domains that I have found do not allow for $p=\infty$ norm control (see, e.g., https://arxiv.org/abs/2110.12967).