What are some elementary bounds for Mertens' Third Theorem?

Question

In particular, I am looking for $A$ and $B$ such that, for all $x>s$, $$\frac{e^{-\gamma}}{\log x} B< \prod_{ p \leq x} \frac{p-1}{p}<\frac{e^{-\gamma}}{\log x}A.$$ Rosser provides both lower and upper bounds, where the former holds for $x>285$ and the latter for $x>1$. I have done research and can not find a lower bound that holds for small $s$, such as $s=1$. Are there any such lower bounds that hold for smaller $s$?