Whisk coconut milk, eggs, 1/4 cup sugar, rum, vanilla, and salt together in a large bowl until thoroughly combined.
Pour 1/2 cup sugar into a heavy skillet over medium-low heat. Cook, stirring constantly, until melted and golden in color, about 8 minutes. Pour into an 8-inch flan or cake pan and swirl to coat the bottom.
Set pan in a larger baking dish. Pour in coconut milk mixture. Pour enough boiling water into the baking dish to make a water bath that comes halfway up the sides of the pan.
Bake in the preheated oven until a knife inserted into the center of the flan comes out clean, 30 to 40 minutes. Remove from water bath.
Refrigerate flan until cold, at least 12 hours. Unmold by gently pushing the sides of the flan away from the pan. Place a serving plate on top of the pan and flip over; tap bottom of the pan firmly to release flan.