Tamron and Sigma make compact 100-400mm telephoto zooms. Both companies also make big 150-600 super telephoto zoom lenses that sell for less than your $1000 cap. For wildlife photography, you can almost never have to much focal length when it comes to zoom lens.
Avoid using a telephoto extender with any of those lenses. Your camera body will lose the ability to autofocus if you use one. If you want the reach, then you want a 150-600mm zoom. When you use an extender, the price you pay for extra focal length is a reduction in the maximum aperture size of the lens you are using. Your f/6.3 lens will become an f/8 lens, and your camera is not able to focus well with such a small aperture, so the camera’s firmware disables AF altogether.
A lens with the focal length you want is a better choice than using an extender for extra reach. But, the 150-600mm lenses are big, heavy beasts. You need to consider how you will safely and securely transport this gear to and from your destination. Professionals use hard cases. When using one you will want to use some means of support, like a monopod or tripod, especially for extended periods of use.
Wildlife photography reminds me a LOT of fishing. You will have the best success by knowing your prey. Knowing when, where, and how to catch the fish, or capture the wildlife in a photograph. Most of the time it takes patience, just like fishing. Sometimes you hit the jackpot, and sometimes you don’t.
If you have ever done much fishing, then you know that the closer that you can get to where the fish are, all the better. This is why people use boats, after all. The same holds true with wildlife photography. Getting closer to the fish is better than trying to cast your line a mile, or using a super jumbo, telephoto lens to get closer to the wildlife.
I will conclude by saying that you will need to practice using a super telephoto lens. It reminds me a lot of looking at the world through a straw. Sometimes it is hard to get your bearings, and figuring out where the subject is relative your viewfinder. Zoom lenses allow you the luxury of zooming out to get your bearings, and then zooming in on the subject. But doing so can mean losing a great shot.
Practice is the key to putting your subject in the viewfinder. This does not even scratch the surface of how would you set up the camera’s auto focus to capture wildlife in action. It wil take hours of practice and experimenting to figure it out.
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"The right mouse button is your friend."
