When an electron passes the double slit it hits the screen leaving a trace but we don't know anything about which one of the two it acctually passed.What about making a single slit on the screen where a stripe appears.We now have a possibility of the electron to pass that slit on the stripe and by placing an additional screen after the first screen we can measure if the pattern on that screen is more wider than it would be if the electron arrived from only one direction from the double slit.If it is wider that would suggest,I think, that the electrons arrived from two directions from the double slit or they pass once from one hole and once from the other with same possibility.It is my rough teory and I ask if it is worth and possible trying this in a lab experiment?I am only concerned if that difference in angles is too small to be acctually measured but what caused me to post this is that if we hit a single slit from two direction the pattern on the screen must be different that the one caused by hitting that single slit from one direction.
I'm afraid that your idea is ingenious but would not shed any light upon the mystery. Whether an electron passes through one of the front sits or both simultaneously, quantum theory would predict that the wave function of any electron passing through the new rear slit would disperse, so when you measured the positions of incident electrons on the additional rearmost screen you would find a spread, and the spread would mask any hint you might have about how the electrons interacted with the original pair of slits.
The question to answer is not whether the stripe is made from one slit or 2, with QM we find they are made from both. But the question to answer is from which slit the electron came without disturbing the pattern, which can not be answered.
So yes the pattern would be broader but the additional slit would introduce more diffraction as well on the final screen.