The reference for this (given in the 3rd edition of Davenport's Multiplicative Number Theory) is:

Grosswald, Émile. "Sur l'ordre de grandeur des différences $\psi(s)-x$ et $\pi(x)-\ell i(x)$." (French) C. R. Acad. Sci. Paris 260 (1965), 3813–3816.

It seems (according to the Math Review) that for $\alpha$ fixed, the error is $O(x^\alpha)$. So, two powers of the logarithm can be saved.

The reference for this (given in the 3rd edition of Davenport's Multiplicative Number Theory) is:

Grosswald, Émile. "Sur l'ordre de grandeur des différences $\psi(x)-x$ et $\pi(x)-\ell i(x)$." (French) C. R. Acad. Sci. Paris 260 (1965), 3813–3816.

It seems (according to the Math Review) that for $\alpha$ fixed, the error is $O(x^\alpha)$. So, two powers of the logarithm can be saved.