  We propose a general formula for higher order corrections to the value of the
Hagedorn temperature of a class of holographic confining gauge theories in the
strong coupling expansion. Inspired by recent proposals in the literature, the
formula combines the sigma-model string expansion with an effective approach.
In particular, it includes the sigma-model contributions to the Hagedorn
temperature at next-to-next-to leading order, which are computed in full
generality. For ${\cal N}=4$ SYM on $S^3$ our result agrees with numerical
field theory estimates with excellent precision. We use the general formula to
predict the value of the Hagedorn temperature for ABJM on $S^2$ and for the
dual of purely RR global $AdS_3$.
