The inequality at the top of page 99 in <A HREF="http://www.math.tifr.res.in/~publ/ln/tifr01.pdf">Lectures on
The Riemann Zeta–Function</A> 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.