Hello, Mary Jo...here is a version from the Farm Journal Country Fair Cookbook. Actually the cake was used for weddings and other special events, and was a midwest favorite. I have seen other versions also, including lemon and chocolate, not sure where. What flavor filling and frosting did your cake have?
I notice it doesn't mention to soften the butter, but would assume that would be a good idea.

Mile-High Butter-milk Cake

4 cups sifted cake flour
1 teaspoon baking soda
1/2 teaspoon baking powder
1/2 teaspoon salt
1 cup butter or regular margarine.
3 cups sugar
1 tablespoon vanilla
2 cups buttermilk
6 egg whites

Creamy Vanilla Filling (recipe follows)

Orange Butter Frosting (recipe follows)

Sift together cake flour, baking soda, baking powder and salt.

Cream together butter and sugar until light and fluffy. Beat in vanilla.

Add dry ingredients alternately with buttermilk, beating well after each addition. Add egg whites; beat at medium speed 2 minutes. Pour batter into 3 waxed-paper-lined 9" round cake pans.

Bake in 350 degree preheated oven for 20 minutes. Reduce heat to 325 degrees and bake 25 more minutes or until cakes test done. Cool in pans on racks 10 minutes. remove from pans; cool on racks.

Spread the top of one layer with half of Cream Vanilla Filling. Place second layer on top. Spread with remaining half of filling. top with third layer. Frost sides and tip of cake with Orange Butter Frosting. Makes 12 servings.

Creamy Vanilla Filling: Prepare 1 (3 1/4 ounce) package vanilla pudding and pie filling according to the package directions for pie filling. cool Well.

Orange Butter Frosting: Combine 5 cups sifted confectioners sugar, 1/2 cup soft butter or regular margarine, 1/4 teaspoon almond extract and 2 to 3 tablespoons orange juice. Beat until smooth.