Kevin Ford proved that if $|t|\geq 3$ and $\frac{1}{2}\leq\sigma\leq 1$, then $$ |\zeta(\sigma+it)|\leq 76.2|t|^{4.45(1-\sigma)^{3/2}}(\log|t|)^{2/3}. $$ This leads to his highly cited explicit version of the Vinogradov--Korobov zero-free region for $\zeta(\sigma+it)$. For $\sigma$ in any known zero-free region for $\zeta(\sigma+it)$, the bound is essentially of size $(\log|t|)^{2/3}$ when $|t|$ is large. When $|t|$ is small, known partial verification of RH will give you much better results.

Kevin Ford proved that if $|t|\geq 3$ and $\frac{1}{2}\leq\sigma\leq 1$, then $$ |\zeta(\sigma+it)|\leq 76.2|t|^{4.45(1-\sigma)^{3/2}}(\log|t|)^{2/3}. $$ This leads to his highly cited explicit version of the Vinogradov--Korobov zero-free region for $\zeta(\sigma+it)$. For $\sigma$ in any known zero-free region for $\zeta(\sigma+it)$, the bound is essentially of size $(\log|t|)^{2/3}$ when $|t|$ is large. Since RH has been verified in the range $|t|\leq 3.0001\times 10^{12}$, in which case one can do much better than a bound of the form $C\log|t|$, Kevin's result is strictly better than $\frac{1}{3}\log|t|$ for $|t|>3\times 10^{12}$, even if you venture all the way to the edge of Kevin's (so far best known) explicit version of the Vinogradov--Korobov zero-free region.