A solenoid which carries a constant current creates a magnetic field. It only generates an electric field if the current is changing. This induced electric field is not the same as the applied electric field which is required to make a current flow in the wires of the solenoid.
A stationary magnet (eg a bar magnet) also generates a magnetic field. The magnetic fields of the solenoid (carrying a constant current) and the stationary bar magnet add up as vectors.
You seem to be complicating the situation by plunging a moving bar magnet into a solenoid which is already carrying an electric current. Textbooks usually describe the solenoid without a current when the bar magnet is plunged into it. The bar magnet then induces an emf in the solenoid, which causes a current to flow in the solenoid, and the current in the solenoid then creates a magnetic field which opposes that of the bar magnet.
During this experiment the magnetic field of the bar magnet does not change, but the magnetic field of the solenoid does change.
Induced electric field in circular wire around solenoid
How is the electric field near a solenoid?
What is the electric field outside a cylindrical solenoid?