The Hover Module allows Sophia 3rd and the derivatives thereof to essentially fly for a very limited time, as fuel is consumed quickly. The flight is of a clumsy nature (tanks aren't very aerodynamic) and a lot of fuel is necessary just to get Sophia 3rd in the air. Once airborne, using short bursts for altitude control is advised to conserve fuel, though a full burn is necessary to reach some areas. Hover can't be used underwater; the Dive Module was built for aquatic travel.
The Hover Module is earned after destroying Photophage in Stage 3. Jason can recover Hover energy by collecting Hover Capsules. All of the Hover energy he's collected to that point will be used to power the Hover Module (which is initially out of power), so collecting Hover capsules before this point is a good idea.
Functions identically to the first adventure.
Elfie makes this upgrade available very late into this adventure. Using Hover quickly consumes the auxiliary battery when in use. This ability is far superior to the Climb (Wall1) ability that the vehicle gets earlier.
The Hover Kit has one important difference here; it can't propel S.O.P.H.I.A. upwards, only slow the vehicle's descent. The Hover Kit consumes the energy gauge, which will refill after a short while of non-activity.