Linnik's theorem for kth prime in the residue class

Question

Linnik's theorm says that for any modulus $m$, the smallest prime in a given residue class mod $m$ cannot be too large: $$ p(a,m)\ll m^L. $$

where $L$ is a constant which has been improved by many authors (5.18 is the best published result, though 5 has been claimed). What can be said about the $k$-th smallest prime $\equiv a\pmod m$? Normally I'd iterate the theorem but all the versions I found require (implicitly or otherwise) $0\le a<m,$ and of course iterating the theorem may give a worse result than can be achieved otherwise.

Answer 1

Corollary $18.8$ of Iwaniec and Kowalski's book states:

Corollary: If $q$ is sufficiently large, and $x\geq q^L$ for an absolute constant $L$, then $$\psi(x,q,a)\gg\frac{x}{\phi(q)\sqrt{q}}.$$

This implies that for $k\ll \frac{q^{L-3/2}}{\log q}$, the $k^{th}$ prime congruent to $a$ modulo $q$ is $$\ll q^L.$$