A while back Ken Thompson wrote: "The success of the 8 inch gun in the Philippines encouraged the US Navy to load up the subsequent battleships with the 8 inch guns as the intermediate battery. Going to the extreme with a 2 story main turret."
Incorrect: The USS Kearsarge & Kentucky, which introduced the 2 story turret, had been authorized back in 1895 & their keels laid in 1896 - well before the Span-Am War. The first generation US BB's, the Indiana class & the Iowa 9completed in 1895 & 96) all carried an intermediate armament of 8 - 8" guns in 4 turrets at the corners of the superstructure. This arrangement however, left little room for secondary guns, 4 - 6" BL on the Indiana class & 6 - 4" QF on the Iowa. The 2 story turret was an attempt to preserve the 4 - 8" intermediate guns on the broadside while providing more room for an effective secondary armament,in the case of the Kearsarge class 14 - 5" QF.
The 2 story would be re-introduced with the 5 ship Virginia class some 5 years latter for much the same reasons, however its many drawbacks meant it would never be tried by any other Navy. In any case the Dreadnought quickly put an end to the use of Intermediate Batteries.