The inequality at the top of page 99 in Lectures on The Riemann Zeta–Function gives $$|\zeta(\sigma+it)|<9t^{1/2},\;\;t\geq 1,\;\;\sigma\geq \tfrac{1}{2}.$$ This still excludes the interval $0\leq t<1$, but at least it's not a large-$t$ result.
Carlo Beenakker
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