The sound quality mainly comes from the audio chip being used. That particular iPod has a chip made by Wolfson, which apparently makes a difference. My personal favorite sound comes from my 3rd gen iPod; it sounds "brighter" to me, compared to later iPods and other Apple devices. I have no idea what audio chip it has.
Anything along the "chain" of the audio signal can have an impact. Impossible to speculate without more info.
If you're going to open up the iPod, here's more important upgrade to consider. Replace the fragile hard drive with solid-state (flash) storage, a compact flash card or SD card on an adapter. This web site has interesting information
https://www.iflash.xyz/
I have not used these particular adapters with 5th gen and later classic iPods, but I have a 3rd gen iPod, 4th gen iPod, and iPod mini modified similarly (using different adapters). They work better now than when they were new, and have more storage. The iPod mini went from 4GB to 64GB! And the adapter plus flash card usually takes up less space, so maybe your "super battery" will fit, without modding the headphones jack.