Shooting from the same point of view will always result in the same image (in terms of magnification) with every 300mm lens you can get (zoom or prime doesnt matter). To achieve a higher magnification your only options are:

• get closer to your subject (your best option but not always possible)
• use a longer focal length like 400mm or 500mm (the second best option if option number one doesnt work)
• use a different camera with a higher megapixel count or with a smaller sensor such you can crop the image tighter (rather spend your money for a longer lens since a new camera is expensive and this would only give you a minor advantage)
• use a teleconverter which will give you a longer focal length (With your lens I strongly recommend you dont. Teleconverters are mostly meant for high end prime lenses and might not work very well)

Now let me explain the best options a little further:

• Getting closer 30% would be equivalent in terms of magnification to using a lens with 30% more focal range. You could try to cut the distance to your subjects in half. If the birds are 100+ yards away getting closer should be quite easy. I often get down to 5 meters to birds using a 500mm lens. There are some good tips on the internet about how to get closer to wildlife. Be respectful, dont run at them and if they are offended then leave. Be quiet and move slowly and you might be lucky. You could also hide yourself using some camo and wait for them if you are up to such stuff.

• 300mm is a relatively short lens for taking pictures of birds. It can certainly work and I did capture some nice images of birds using a 300mm lens but I would suggest you to look for something like a 150-600mm lens. Those are more or less affordable and they feature some nice reach.