Not surprisingly, such questions date back to the original Stern-Gerlach experiments. An early experiment, perhaps not exactly what you seek, is reported by TE Phipps and O Stern (yes, the same Stern) in Zeitschrift fur Physik 73(3-4) 185-191 (1932). Another, more recent article, is K. Brodsky et al., Europhysics Letters 44(2) 137-143 (1998) on 'Single and double interaction zone with comoving fields in Stern-Gerlach atom interferometry' - this is probably way more sophisticated that you seek, but shows that double Stern-Gerlach experiments are not uncommon, but perhaps not called that.
A nice article that covers most of the related theory in an accessible way is AR Mackintosh, 'The Stern-Gerlach experiment, electron spin and intermediate quantum mechanics', Eur. J. Phys 4 97-106 (1983). This is theory only.
Don't think of magnets and electrons, just think simpler.
Let's play some ice hockey. To simplify this game, it's just one player, who I'll call "us", with a puck, in a really big rink. There isn't even a proper enemy player, just a big wall in the middle of the rink, trying to stop us. We can only go through a "gap" in the wall. Let's say the puck is moving forward, straight from our goal to the target goal, and we're travelling with it.
We tap it once, say to the left, so that it can go with us towards the gap in the wall.
Then when it's in line with the gap, we tap it once to the right, so that it goes straight. We thereby pass through the opening in the wall with the puck.
Now we tap it again to the right, so that it comes to the center line again. When it's in line with the center line, we tap it again to the left to stop it. That's what this last magnet is doing, it's giving this last tap to stop the things.
The pattern of taps is L, R, R, L. If we are really precise with our taps, or the "wall" is not very big, we can condense the two R taps into one bigger tap 2R, which does both of them. Instead of going straight through the gap it now goes diagonally through the gap, possibly turning right as it goes through, depending on when we tap it.
Let's say the Y direction measures progress towards the other net, with the X direction perpendicular. The puck's horizontal momentum $p_y$ is some constant because there is no friction. Its horizontal momentum $p_x$ however changes as we tap it: it starts at $0$, then goes to $p_1$, then goes to some $-p_2$, then goes to $0$ again. This requires at least three impulses: one of magnitude $p_1$ to take it from $0$ to $p_1$, one of magnitude $-p_1 - p_2$, to take it from $p_1$ to $-p_2$, and one of magnitude $+p_2$, to take it back to $0$.
Best Answer
The evolution for a particle with a well defined spin along the $z$-axis is : $$\begin{array}c I && III && V \\ |\phi_I\rangle\otimes|+\rangle_z & \longrightarrow & |\psi_\uparrow\rangle\otimes |+\rangle_z & \longrightarrow & |\psi_V\rangle\otimes|+\rangle_z \\ |\phi_I\rangle\otimes|-\rangle_z & \longrightarrow & |\psi_\downarrow\rangle\otimes |-\rangle_z & \longrightarrow & |\psi_V\rangle\otimes|-\rangle_z \\ \end{array}$$ where $|\phi_I\rangle, |\phi_V\rangle$ are wave-packets localized in region I and V respectively, and $|\psi_\uparrow\rangle$, $|\psi_\downarrow\rangle$ are localized in region III on the up and down path respectively.
In the usual Stern-Gerlach experiment, the position of the particle in region III is measured, which allows to tell $|\psi_\uparrow\rangle$ from $|\psi_\downarrow\rangle$ and in turns gives us the spin of the particle.
Now, consider any particle at the start of the experiment. Its state is : $$|\Psi_I\rangle = |\phi_I\rangle \otimes |u\rangle$$ where $|u\rangle =\alpha |+\rangle_z + \beta|-\rangle_z$ with $|\alpha|^2 +|\beta|^2= 1$. By linearity, it evolves as : $$\begin{array}c I && III && V \\ |\phi_I\rangle \otimes |u\rangle & \longrightarrow & \alpha |\psi_\uparrow\rangle\otimes |+\rangle_z + \beta |\psi_\downarrow\rangle\otimes |-\rangle_z & \longrightarrow &|\psi_V\rangle\otimes |u\rangle \end{array}$$
ie, the spin of the particle is unchanged. Looking only at $III\rightarrow V$, the superposition interferes just in the right way to give the initial spin state.
(In the book, they choose $\alpha = \beta = 1/\sqrt 2$, ie $|u\rangle = |+\rangle_x$, but this works for any initial spin state)