A solid object is constructed from a right circular cone whose base has a radius of $3$ inches and height $6$ inches. A smaller cone with height $2$ inches is removed from its top, parallel to the base, forming a frustum. A new, smaller cone is then placed on the top surface of this frustum, such that its base is the top surface of the frustum and its height is $3$ inches. What is the total volume of the resulting solid, in cubic inches?