Although this is almost an engineering question, there are some simple physical constraints that mean the answer is an emphatic no.
At staging, significant change in momentum / trajectory of the payload-bearing stages can only arise from high impulse transfer between the payload bearing and jettisoned stage; this statement is a reformulation of conservation of momentum.
High impulse transfer implies extended interaction between the jettisoned and payload bearing stages. But the main purpose of staging is clean separation. You want to get rid of the jettisoned stage with minimal risk of collision with the payload bearing stages, in particular with its precious and delicate rocket nozzles. Without considerable added complexity (e.g. some kind of sheathing arrangement whereby the jettisoned stage can be smoothly accelerated relative to the load bearing stages whilst the two still remain in contact as in a rail gun like arrangement) and therefore added weight, the goal of clean, low risk separation is incompatible with the goal of extended contact / interaction between the two parts for significant impulse transfer.
Otherwise put, to get significant thrust, by the the Tsiolkovsky equation one needs to jettison mass at high exhaust velocity. The only kind of matter that can be accelerated to high exhaust velocity in the confined spaces of a rocket system seems to be gas in the combustion chamber.
We can do a simple calculation to show that the quest for further impulse from the spent stage is likely to be highly costly for very meagre returns. Let's take, for example, the mighty S1-C stage of the Saturn V launch vehicle. Its fully laden weight was about 2200 tonnes, whilst its spent weight was around 110 tonnes (the 2200 tonnes seems readily substantiated by a Google search, I'm having a little trouble finding the source for the 110 tonnes - I would appreciate a link if someone has one). This 96% fuel to container ratio is typical of rocket systems (see the Don Petit article below). The fuel exhaust speed was about 2500 meters per second. The most sophisticated railgun arrangement might accelerate the 100 tonne spent stage to maybe hundreds of meters a second relative to the payload bearing stages, i.e. an order of magnitude slower than the exhaust velocity. Therefore, the impulse available from the spent stage in an extremely optimistic scenario is about $0.04 \times 0.1$ or of the order of one half of one percent of the impulse won from burning the fuel. Therefore, there is little room for added sophistication before the chasing of this further impulse becomes a futile and self defeating exercise. When one sees a rocket on the launchpad, it looks like a mighty and strong metal structure, so one is tricked into believing that the jettisoning of the spent structure at even a low speed would yield a great deal of impulse. In reality, however, the rocket is the flimsiest of skins, much weaker in relation to the mass of fluid that is must contain than a child's balloon and barely strong enough to contain a mighty column of liquid that must be lifted into the sky. The rocket's mass in comparison with the impulse yielding fuel is trivial.
One could imagine some group of engineers in the future working out how the added impulse from the jettisoned stage might replace, say, the need for ullage motors on the load bearing stage to accelerate the next stage's freefalling fluid and bring it into contact the fuel pumps. Clever design might just see some significant, but small, task like this being taken over by your idea. But the gain of significant further impulse is all but impossible.
As a companion exercise to your consideration of this problem, I think you should read:
Don Petit, "The Tyranny of the Rocket Equation"
which gives a deep feeling for how barely our technology lets a meagre handful of us escape the gravity of a large rocky planet like the Earth. After 60 years of space flight, there is little evidence that we have improved the reliability of rockets; they are so close to the limits of our technological ability that any complexity increase with any added weight whatsoever is impossible. Explosives to throw the spent stage away are a simple, low risk and light solution; unfortunately they are, well, kind of explosive and thus the interaction between the two stages is short lived with minimal momentum transfer. It seems unlikely that simple and reliable explosive stage separation will be superseded in the foreseeable future.
