I have a couple of questions regarding possible improvements of the Reidemeister-Schreier index formula: let $G$ be a $d$-generated group and let $H$ be a subgroup of $G$, then

$$d(H) \le (d-1) \vert G:H\vert + 1.$$

Therefore, in a finitely generated group $G$, the minimal number of generators of a subgroup is bounded above by a "linear" function of its index in $G$.

In general this bound might be too big for particular classes of groups. For example in "Subgroup Growth" Theorem 11.9, it is shown that if $G$ is a PSG profinite group, then there exist a constant $C>0$ such that

$$ d(H) \le C \cdot \sqrt{\log|G:H|}$$

for every $H\le_o G$.

My questions are the following:

are there other known improvements of the Reidemeister-Schreier index formula for particular classes of groups?

Is the bound known to be "logarithmic" for other classes of groups? (By logarithmic I mean of the form $a\cdot(\log\vert G:H\vert)^b$ for constants $a,b>0$.)

I am particularly interested in the classes of finite perfect monolithic groups and profinite groups of slow subgroup growth (for example $n^{\log n}$).

Any comments or references would be very welcome. Thank you in advance.