  We give a new proof of the fact that the growth (with respect to $n$) of the
$p$-adic valuation of $h_{n,2}^{-}$ is linear, where $h_{n,2}^{-}$ denotes the
minus part of the $(S,\{2\})$-refined class number of the cyclotomic field
$\mathbb{Q}(\mu_{p^{n+1}})$, as defined by Hu and Kim in [J. of Number Theory
158 (2016), 73-89]. As a consequence of our proof, we obtain an explicit
relation between the $p$-adic valuation of $h_{n,2}^{-}$ and the $p$-adic
valuation of $h_{n}^{-}$, the minus part of the class number $h_n$ of the
cyclotomic field $\mathbb{Q}(\mu_{p^{n+1}})$.
