Heat oil over med heat in a deep 12-in skillet that has a lid. Meanwhile, scrub the potatoes, and either peel them (if you wish) or leave the peel on. Cut them into 3/4” cubes, adding them to the pan as you cut. Raise heat to high, then cover and cook until the potatoes are partially browned, stirring occasionally. While the potatoes cook, peel & mince the garlic, and set it aside.
When the potatoes are partly browned and starting to be tender, add the sausage to skillet, breaking it up into clumps. Cook until potatoes are golden and sausage has browned, 8–10 minutes or so. Stir often, so that the sausage crumbles into bite-sized pieces.
Add the minced garlic and the rosemary to the pan. Stir well to distribute, and then add the diced tomatoes with any juice in the can. Cook the hash until half of the liquid evaporates, and the hash comes together: 2-3 minutes more.
Serve hot. You might like to top this with an egg, or pair it with toast, or even roll it up in a flour tortilla. Refrigerate any leftovers, and be prepared to fight for your share as a midnight snack.