In short, no. The reason is, the hardest part by far of getting to space is getting to Low Earth Orbit. As the saying goes, once you've done that, you're half way to anywhere.
It would be difficult to make the elevator stay put to non-anchored locations. Furthermore, it would mess with the speed required to get to each of these locations, and in the end, wouldn't be very practical.
Given this, the most practical thing that could be done along these lines would be to build a very large platform on top of a mountain, and raise the object to the top of the platform to launch. That gives very little theoretical benefit, although it does cut down on wind resistance and such.
There is, however, a system that could help once you have obtained orbit, using magnetic tethers. I won't bother to explain all of the physics, but I will point you to Space TethersSpace Tethers, which discusses it further, and also provide a graphic from their website which demonstrates it further.