For people who are seeking the second pulse / Freq. doubling effect I can tell you one thing:
You will not see a second pulse alone, but an extend of the "pulse on" in the falling direction, because the inductor(s) trying to maintain the current flow and the stored energy in the magnetic field is colapsing, if you turn on before in colapses you get the "capacitance charging effect".
another hint:
(https://i.postimg.cc/MHmL0nF0/injector.gif)