[Heath-Brown (2016)][1] proved that
$$\zeta(\sigma+it)\ll_\varepsilon t^{\frac{1}{2}(1-\sigma)^{3/2}+\varepsilon},\qquad 0\leq\sigma\leq 1,\qquad t\geq 1.$$
The exponent $1/2$ can be improved to $\frac{8}{63}\sqrt{15}=0.4918\dots$ for $1/2\leq\sigma\leq 1$. Moreover, for any $\lambda>\frac{2}{\sqrt{27}}=0.3849\dots$, there exists $\sigma(\lambda)<1$ such the exponent $1/2$ can be improved to $\lambda$ for $\sigma(\lambda)\leq\sigma\leq 1$.

  [1]: https://arxiv.org/abs/1601.04493