After the field coils have been de-energized, a residual field will remain and decay.
They are similar to an electro-magnet in this regard.
As magnets produce their own magnetic field, I suspect that warp field coils naturally produce their own subspace field, which the electro-plasma amplifies.
Warp drive is a non-Newtonian propulsion method. TNG Technical Manual page 54.
Reaction products are channelled into field coils, producing layers of energy that push on each other, pushing the ship into subspace and propelling the ship faster than light.
According to the TNG Technical Manual page 28, "Decaying warp field energy surrounding the saucer module is managed by the driver coil segments of the impulse engines. This energy will take, on average, two minutes to dissipate and bring the vehicle to its original sublight velocity."
Subspace also imparts drag upon a ship, one of the reasons ships are built with sweeping curves. Subspace though like space, is also a medium containing layers. Each successive layer can require thousands of times the energy needed to travel in a previous layer.