Puree the mango in a high-powered blender or food processor. Remove the puree from the blender. Most of it will be set aside for layering the pudding.
In a medium bowl, whisk together 1 tablespoon of the pureed mango along with the coconut milk, water, maple syrup, and vanilla extract. Slowly pour in the chia seeds, whisking constantly.
Cover the bowl, and transfer the mixture to the refrigerator. Refrigerate for an hour to let the pudding thicken. After the hour, stir the pudding and see if the thickness is an appropriate pudding texture. If you use full-fat canned coconut milk, it may be too thick. In this case, you can stir in a couple tablespoons of water to get the pudding to the right texture.
Divide the pudding into 4 jars. Layer chia pudding at the bottom of each jar, followed by mango puree, then more pudding, then a dollop of mango puree on top.
Leftover pudding can be stored in the refrigerator for several days.