Correct, there's no way out of the box to reorder fields on the fly in the Entry editor. If you're comfortable with Craft plugin development though I'm sure it's possible to extend Craft and add the ability to do what you want to do. It's also possible that someone has already developed a plugin that achieves this but I personally don't know of any off the top of my head.
However, it seems in your case that if you want to be able to rearrange fields in the Entry editor, your best solution is to put those fields inside a Matrix field. When you have created several fields inside a Matrix field, those "sub-fields" are then arrangeable in whichever order you prefer in the Entry editor. If you've never used Matrix fields before, here's the documentation link so you can read up on it: https://craftcms.com/docs/matrix-fields
Another alternative would be to use the Neo field - if you decide to use Neo, you can just reference your currently existing fields and then add only the Neo field to the page.
Hopefully that helps to answer your question. Cheers!