Heat 2 tablespoons oil in heavy large pot over medium-high heat.
Add sausage; cook until brown, stirring often, about 8 minutes. Transfer sausage to paper towels to drain.
Add onions and garlic to pot and cook until translucent, stirring often, about 5 minutes.
Add all potatoes and cook until beginning to soften, stirring often, about 12 minutes. Add broth; bring to boil, scraping up browned bits. Reduce heat to medium-low, cover, and simmer until potatoes are soft, stirring occasionally, about 20 minutes.
Using potato masher, mash some of potatoes in pot. Add browned sausage to soup. tablespoon oil.
Stir in spinach and simmer just until wilted, about 5 minutes.
Turn off the heat.
Stir in remaining 1 Tbsp of olive oil to allow the fresh flavor to come through.