Preheat oven to 325 degrees F. Grease and flour a standard size Bundt pan or spray generously with a good flour-added baking spray.
Mix the cake flour, sugar, baking powder and salt in bowl of a stand mixer. Add chunks of softened butter and 3 tablespoons of the vegetable oil. Continue mixing on medium low speed until mixture looks like fine gravel. Add the pudding mix, milk, eggs, remaining oil, rum and vanilla. Continue beating with the paddle, raising speed to medium. Beat until well mixed, scraping bowl often. Batter should be very smooth.
Sprinkle the chopped nuts across the bottom of the pan. Pour batter over nuts and bake for 55 minutes or until a cake tester comes out clean and cake springs back.
Prepare Rum Soaking Glaze. Combine butter, water and sugar in a small saucepan and bring to a boil. Reduce to a simmer and cook until sugar is dissolved and syrup is well combined and a little thicker.
Remove from the heat and stir in the rum.
While cake is still cooling upside down in the pan, pour some of the hot syrup on top of the cake (I poked holes all over it with a toothpick, allowing it time to soak in (this may take a few minutes as there will be a lot of syrup) continue to add syrup until all of the syrup is added. Allow cake to cool completely in pan before turning out onto serving platter.