Black Rice Pudding

Bring rice, 3 cups water, and 1/4 teaspoon salt to a boil in a 3- to 4-quart heavy saucepan, then reduce heat to low and simmer, covered with a tight-fitting lid, 45 minutes (rice will be cooked but still wet). Stir in sugar, a scant 1/4 teaspoon salt, and 1 1/2 cups coconut milk and bring to a boil over high heat, then reduce heat to low and simmer, uncovered, stirring occasionally, until mixture is thick and rice is tender but still slightly chewy, about 30 minutes.
Remove from heat and cool to warm or room temperature, stirring occasionally, at least 30 minutes. Just before serving, stir pudding and divide among 8 bowls. Stir remaining coconut milk and drizzle over pudding.