Keep in mind that you may have put down too much seed next to the driveway and that is why it is so lush. It won't continue to look like that as the grass chokes itself out. As far as the rest of the lawn goes I'd tend to be sure I am fertilizing correctly and doing the other things correctly before adding more seed. Have you ever planted one grass seed and let it grow as much as it can? The resulting plant is large. Point being to make the most of what you have before you start overseeding. If you have a bluegrass lawn it will spread nicely and thicken up when treated properly, but any lawn should thicken up if it has been malnourished.

Here is something that might help. I look for any excuse to use this picture from 2010.

That is a zoysia lawn fertilized in mid May and photographed in mid June. The fertilizer used was alfalfa pellets (rabbit chow). It is clear that the fertilized area is more dense, taller, and deeper green. This is pretty typical for grasses fertilized with organic fertilizers. Thanks to mrmumbles for posting the picture.