We show that in the random hyperbolic graph model as formalized by Gugelmann, Panagiotou, and Peter (2012) in the most interesting range of 1,2 < α < 1 the size of the second largest component is Θ((log n)1/(1-α)). Our research is motivated by the question raised by Bode, Fountoulakis, and Müller (2013) regarding the uniqueness of linear size components in random hyperbolic graphs, which naturally leads to the question regarding the size of the second largest component. We also show that for α = 2 with constant probability the corresponding size is Θ(log n), whereas for α = 1 it is Ω(nδ) for some δ > 0.