What would happen if you add a second double-slit behind the first in the double-slit experiment?

Dude that video is really, really misleading.

Firstly, particles are particles, but they have a "wavefunction" which is a function of the probability of certain momentum values, and it can be seen as a function of how localized the particle is. Basically, the wavefunction is a function of the particle's existence. The wave is still considered to be physically real because the wavefunction is a function of space.

Secondly, the reason why you don't get the interference pattern when you measure the system is because in order to measure the system, you have to mix two systems together and this screws up the experiment. It causes the wavefunction, which is a complex exponential harmonic oscillator function, to collapse into a Dirac delta function, so the interference pattern is gone. It's called a "wavefunction collapse".

What happens is that the electron wave (which is spread across the whole area) goes through both slits and turns into two waves, then the two wavefunctions of the two electron waves multiply eachother and that results in an interference pattern.

To answer your question though, this is what would probably happen...

1) The electron wave goes through the first pair of slits, splits into two waves, and then interferes with itself. If the two waves have functions A and B, then the interference pattern is just AB.

2) Now the interference wave "AB" goes through the second pair of slits, and splits into two different functions "D" and "E". The final interference pattern would be DE.

If you measure the first slit only, you still mix two systems together, so the wavefunction still collapses. Even the most subtle measurements will cause the wavefunction to collapse, there's no way to get around it.