Do the self-induction cause change in magnetic field which again causes a self induction?
At first: The descriptions A cause B and B cause C does not mean any time difference. A, B and C occurs at the same time when we talk about electric and magnetic fields. That clear? I hope so and continue.

If you look at Maxwell equations about what is caused by the changes of the field strengths, it really is as you commented to user Felthry: Non-zero inductive emf really causes a magnetic field except in one special case. That case is: The current grows or decreases at constant speed (A/s), has allways done so and will continue it forever. In other cases there exists a radio wave.

In practical coils we very likely use so slowly changing currents that the second order effect "the magnetic field that is caused by the induced emf" can be ignored without a big error. Actually we ignore the radio wave that occurs around the inductor and spreads to the space.

When it can be ignored? As long as the dimensions of the coil are so small that all changes occur practically at the same time in the whole coil. Increase the frequency or make the coil bigger, the error grows. There's no strict limit, when coils become useless. One adage or maxim about it is "keep the maximum dimensions of your circuit equal or less than 2% of the wavelength, then you can ignore the in-circuit waves and get usable results by calculating currents and voltages.