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

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 $n \geq 19913559, \; \; \;$ $s(n) < e^\gamma \log \log n.$

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 $n \geq 19913559, \; \; \;$ $s(n) < e^\gamma \log \log n.$