A minor CSS change you can make within a theme is to prioritize the tabs on the left to put the custom group tab first… this looks like:
.emoji-picker__sections-nav [data-section="default"] {
order: -1;
}
(“default” being the emoji group name as seen in /admin/config/emoji)
So that makes it easier to click the button on the left to jump to the custom emoji.
Beyond this, there’s not a great way to adjust the ordering of the sections themselves at the moment.
The emoji in the menu are lazy loaded on scroll, so even if you adjust the content ordering with CSS… the custom emoji wouldn’t be loaded when the menu is first opened.
We could potentially add a hook to customize the order in a theme if it’s something more people would generally find useful.