Is there a way to create a new icon subset and put custom icons into it? Or, to put custom icons into the existing subsets?
No, there’s no way to categorize or organize in another way your custom icons. However, your icons will show in the Quick Add search results, so you can just use Quick Add to get immediate access to them (you can use the keyboard shortcut / or + to send focus to the search box and just start typing). If you did that, you could put something in the icon name to help with searching.
We have plans for organization down the road, but nothing coming right away. Sorry I don’t have a better answer for you yet!