How fast does the proportion guaranteed by dirichlet converge?

Question

I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in NT. What I'm using is that for prime $q$ $$\frac{\{p\equiv1\mod q,\; p\leq x\}}{\pi(x)}=\frac1{q-1}+o(1)$$ can I say anything about this $o(1)$ term? Specifically, I'm hoping that $$\frac{\{p\equiv1\mod q,\; p\leq x\}}{\pi(x)}-\frac1{q-1}\ll \frac{\log x\log\log x}x$$ although any kind of result better than just $o(1)$ would help