  High-temperature ($q\to1$) asymptotics of 4d superconformal indices of
Lagrangian theories have been recently analyzed up to exponentially suppressed
corrections. Here we use RG-inspired tools to extend the analysis to the
exponentially suppressed terms in the context of Schur indices of $N=2$ SCFTs.
In particular, our approach explains the curious patterns of logarithms
(polynomials in $1/\log q$) found by Dedushenko and Fluder in their numerical
study of the high-temperature expansion of rank-$1$ theories. We also
demonstrate compatibility of our results with the conjecture of Beem and
Rastelli that Schur indices satisfy finite-order, possibly twisted, modular
linear differential equations (MLDEs), and discuss the interplay between our
approach and the MLDE approach to the high-temperature expansion. The
expansions for $q$ near roots of unity are also treated. A byproduct of our
analysis is a proof (for Lagrangian theories) of rationality of the conformal
dimensions of all characters of the associated VOA, that mix with the Schur
index under modular transformations.
