Preheat the oven to 350°F. Place a rack in the center.
Toast pecans on a sheet pan for 6–8 minutes until fragrant. Cool briefly.
Set the pie shell on a foil-lined sheet. Scatter pecans and shredded coconut in the shell.
Whisk brown sugar, granulated sugar, salt, and cinnamon in a bowl to break up lumps.
Whisk in beaten eggs, corn syrup, coconut milk, melted butter, vanilla, and rum until smooth.
Pour filling evenly over the pecans and coconut in the pie shell.
Shield the crust edge with foil. Bake 45 minutes, then remove foil and bake 8–12 minutes more.
Remove when the center jiggles slightly and the top is set. Cool at least 2 hours before slicing.