I would go for an upgraded walkabout lens, something like a Sigma 17-50mm or older 17-70mm. I use the 17-70mm and noticed I was getting much better results than my kit lens and I started to really see the quality of my photography.
For further range, if you can stretch to an old 70-200 or 80-200 F2.8, it will blow the socks off your 70-300.
If budget wont stretch, have a look at the Sigma 50-150mm F2.8. On your Nikon, this is equivalent to a 70-200. Its lightweight, sharp and very versatile.
I think you need to be looking at lenses F2.8 to F4. Speed and performance in low light will give you the improvements you're looking for.