Is there an easier way to move a pet from one stable tab to another, other than just moving them to a different stable, filling up the one tab, and then moving it back to the first stable?

I'm not sure what you mean. If you want to move pets within the same stable you just click it and then move it to the desired slot. If you want to move pets from stable to stable you click the pet and then click the desired stable. If you're moving a lot of pets move them all to the stable and then click the stable you moved them to and then move them to the desired slots. There is not an easier way at this time.

I've had luck with the method you are talking about, but if I have a pet in stable 1 tab 1, and I want that pet in stable 1 tab 2, is there a way to get that pet there without moving it to stable 2, filling up stable 1 tab 1, and then moving it back to stable 1 so that it ends up in the second tab.

O_o

Err...all you do to move a pet to another tab is select the pet and then click on the tab you want it to be in and then click on an empty spot.

you do not need to fill tabs and move pets from stable to stable just to move a pet from one tab to the other.
Pets can be moved from tab a to tab b by:
1. clicking the pet that you would like to move
2. clicking to view the tab where you would like it to be
3. clicking the slot you would like that pet to be in.

e.g. I want to move my pet Nami from stable 1, tab 1 to stable 1 tab 2. I would click Nami's slot in tab 1. click to open stable 1, tab 2 then click an available slot in tab 2.

Oh, thanks. I didn't realize that the pet was saved to the cursor so to speak. I thought it was only letting me switch tabs when I clicked on the tab. Thanks so much.