The secondary hull, sometimes referred to as the stardrive section, engineering hull or battle section, was a structure that formed part of a starship. It was usually located to the rear and below a ship's primary hull, and was either directly attached, or separated by a connecting dorsal.
In Federation starships, the navigational deflector, engineering, battle bridge, cargo and shuttle bays were generally located in the secondary hull. (TOS reference: Mr Scott's Guide to the Enterprise; TNG references: Star Trek: The Next Generation Technical Manual, Star Trek: The Next Generation Sketchbook: The Movies; TLE novel: The Buried Age)
The secondary hull is encompassed by the aft or secondary warp field lobe generated by the warp nacelles. The concave aft hull undercut found in many Federation starships prevents 'pinwheeling' as it allows for varying degrees of field flow attachment. (Ships of the Line 2011; TNG reference: Star Trek: The Next Generation Technical Manual)
Around the turn of the 2160s decade, some NX-class starships underwent a refit which, among other things, added an engineering hull. This new hull housed the relocated engineering section including an upgraded warp core and related systems, and a more powerful navigational deflector. At the insistence of Admiral Jonathan Archer, the ships were redesignated the Columbia-class. (Ships of the Line 2011; ENT - Rise of the Federation novel: A Choice of Futures)
In the mid-23rd century, the Constitution-class was capable of saucer separation, a maneuver that separated the primary and secondary hulls enabling each section to have autonomous flight capability. However, this procedure was to be used only in emergencies as the two sections had to be reconnected at a starbase or other similar facility. (TOS episode: "The Apple"; TOS comic: "'Til Death"; TOS - Unlimited comic: "None But the Brave")
From the late 23rd century and through the 24th century, as more starship classes were designed with the capability of hull separation, a secondary bridge, or battle bridge, was incorporated within the secondary hull. These became more prevalent as some classes, such as the refit Excelsior-class and Galaxy-class, were capable of autonomous reconnection without external support. (TNG movie: Generations; TNG episode & novelization: Encounter at Farpoint)