Choie et. al. had more concerns than just  odd $n;$  if we ask when their argument kicks in, it is simpler in appearance.
Small prime $p,$

$$s(n)=s(pn)/s(p)< \frac{p}{1+p}\left(e^{\gamma}\log(\log(pn))+\frac{0.64821365}{\log(\log(pn))}\right) \; ?< ? \; e^{\gamma}\log(\log(n))$$

or
$$   \frac{p}{1+p}\left(  \frac{\log(\log(pn))}{
\log \log n}+\frac{0.363945701}{\log(\log(pn))\log \log n}\right) \; ?< 1 ?        $$

This decreases as $n$ increases, using simple
$$  \log \log n < \log \log pn  < \log \log n +  \frac{\log p}{\log n}  $$

$$  \bigcirc \bigcirc  \bigcirc  \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc     \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc  \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc $$



For odd $n \geq 17,$ we find $s(n) < e^\gamma \log \log n.$  

For $n \neq 0 \pmod 3$ and $n \geq 56, \; \;$  $s(n) < e^\gamma \log \log n.$ 

For $n \neq 0 \pmod 5$ and $n \geq 898, \; \; \;$  $s(n) < e^\gamma \log \log n.$

For $n \neq 0 \pmod 7$ and $n \geq 19479, \; \; \;$  $s(n) < e^\gamma \log \log n.$

For $n \neq 0 \pmod {11}$ and $n \geq 19913559, \; \; \;$  $s(n) < e^\gamma \log \log n.$