It's easier if you're using the button overlay. Running at the cat and spamming one of the action buttons (I forget which one) will give your pet a hint, and it'll move half a tile away.
Also, this is the method to use on items you've placed, but can't be put in your inventory. Example: Chests that already have items inside can be shoved into neighboring empty tiles.
It took me an embarrassingly long time to figure this out.
