We show that in the random hyperbolic graph model as formalized by Gugelmann et al. in the most interesting range of $\frac12 < α< 1$ the size of the second largest component is $Θ((\log n)^{1/(1-α)})$, thus answering a question of Bode et al. We also show that for $α=\frac12$ with constant probability the corresponding size is $Θ(\log n)$, whereas for $α=1$ it is $Ω(n^{b})$ for some $b > 0$.