Rinse the rice in three changes of cold water, or until the water becomes only slightly cloudy from the rice. Drain well in a sieve.
Heat the ginger with the oil in a small (2-qt.) saucepan over medium-high heat until it begins to sizzle steadily and becomes fragrant, 1 to 2 minutes. Add the rice and cook, stirring, until the grains and ginger start to brown in places, about 2 minutes. Stir in the coconut milk, 1-3/4 cups water, and the salt. Bring to a boil and then reduce to a simmer. Cook until the liquid has reduced to about the same level as the top of the rice, 5 to 7 minutes. Cover, reduce the heat to low, and cook without disturbing the rice until the liquid is absorbed and the rice is tender, 15 minutes.