Rinse the clams several times under cold running water. Transfer to a large pot and add 3 cups water. Bring to a simmer over medium-high heat, then cover and cook until the clams open, about 6 minutes. Strain through a fine-mesh sieve into a large bowl; reserve the broth. Transfer the clams to a bowl and cover with plastic wrap so they don't dry out.
Cook the bacon in a large pot over medium heat until crisp, about 10 minutes; remove with a slotted spoon and drain on paper towels. Melt the butter in the same pot over medium heat. Add the celery, onion and thyme and cook, stirring occasionally, until soft, about 10 minutes. Add the flour and cook, stirring, 1 minute. Stir in the reserved clam broth, 1 1/2 cups water and the potato. Bring to a simmer and cook until the potato is tender and the soup thickens slightly, about 15 minutes.
Meanwhile, remove the clams from their shells and roughly chop.
Reduce the heat under the soup to medium low. Whisk in the half-and-half and warm through (do not boil). Remove from the heat. Stir in the chopped clams; season with salt and pepper. Ladle the soup into bowls and top with the reserved bacon and chives.