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