With your cursor on the method definition line (you do not have to and press ctrl+shift+up or ctrl+shift+down, to move up or down respectively.
You can also to ctrl+shift+numberpad - to quickly collapse everything so you can focus on moving around (plain - works on my laptop as well, not sure why) and ctrl+shift+numberpad + to get back to see everything (ctrl-shift-equals works on my laptop as well).
Not a perfect answer yet, due to a bug in IntelliJ.
Though IntelliJ offers this feature implicitly, but it needs to be enabled as well as fixed. The OP's suggested way is technically arranging methods in depth-first order. However, if you use Breadth-first ordering(which works properly), it should reduce the manual work of moving functions by a lot, by arranging all caller and callee methods together.
IntelliJ has a built in system that allows you to specify how to order your methods. You need to go to Settings (Ctrl + Alt +S) -> Editor -> Code Style -> Java -> Arrangement (tab) and scroll down until you find the icons with methods. There you can manipulate the options to sort them by visibility, or alphabetically, or to keep related ones grouped together.
Here is a screenshot of my settings which will order methods automatically by visibility (public, protected, private) and alphabetically (a-z).
The blue highlights show the currently selected rules.