In a large pot, heat olive oil over medium heat. Add the chopped onion and garlic, sautéing until translucent, about 3-4 minutes. Add the carrots and celery to the pot, cooking for an additional 5 minutes until they begin to soften.
Pour in the liquid (chicken broth, beef broth, or reserved cooking liquid) along with 2 cups of water. Add the diced potatoes, thyme, salt, and pepper. Bring the mixture to a boil, then reduce heat to low and simmer for 15 minutes or until the potatoes are tender.
Stir in the chopped leftover corned beef, chopped cabbage, and continue simmering for an additional 10 minutes to allow the flavors to meld together and cabbage to get as tender as you'd like. Taste and adjust seasoning if needed