When potatoes are fully softened, blend soup until mostly smooth but a little texture remains using a hand blender directly in the pot or by transferring to a countertop blender in batches. (seriouseats.com)
Puree the soup using a stick blender (or in a regular blender in batches, but take care and leave room for steam to escape). (homespuncapers.com)
When the soup is done cooking, use an immersion blender to blend the ingredients until smooth (if you don't have an immersion blender you can transfer the soup to a normal blender in batches and blend until smooth). (emilybites.com)