I do it the old-fashioned way - with a couple twists:
- create preset in Axe-Edit
- tweak and save to the next patch up by pressing 'save' on Axe face, and use scroll wheel to put it above (or below, if I've already got patches above) original patch
Occasionally, I use the 'save to another location' in Axe-Edit, but it's more work when doing a bunch of them.
When I get enough patches to where I'm not making any ground, I consolidate to a few variants (maybe only one). After doing this with several amp types, I've taken up thirty or forty patch spaces, and then I consolidate further, down to two or three amp types. During this, I rename the junked patches to 'null'. Then, as I did a bunch last night, I manually move each patch down by going to the first one in Axe-Edit, pressing save on the panel, scrolling down with the wheel, and pressing save to finalize. Then, in Axe-Edit, I rename the old patch place to 'null', press the save button on the screen, and press the 'up patch' button to begin the process over again.
Axe-Edit stays on the patch it's displaying, despite manual (from the unit face) change on the Axe; whereas the Axe will reflect real-time changes from Axe-Edit; so they kind of overlap.
I'd love if there were drag-and-drop, and shift and ctrl grouping functions in Axe-Edit (like in Explorer), as I could move a patch(es) around within the patch selection window without all the above work. Also, I could copy selected bits of a signal chain and paste them to another patch.
** Note that I play mostly high gain, so cleans are far fewer and will ultimately be down to one or two. Names I haven't gotten much past the amp model names, because I still have too many. I may end up having the same amp type for different guitars, but each will likely be slightly different. Ultimately, I hope to have one clean amp type, and one or two gain amp types.