Just to lay some groundwork (I'm sure most will already know this, but maybe some don't) the first number of a binocular configuration is the magnification, the second number is the objective lens size. So an 8x42 is 8x magnification, with 42mm objective lenses. The exit pupils are the bright circles you see in the eyepieces when you hold the binoculars away from your face. You can calculate exit pupil size by dividing the objective lens diameter by the magnification. So for a 8x42 binocular, 42÷8 gives you a 5.25mm exit pupil. For hunting crepuscular animals I would recommend something with at least a 5mm exit pupil.
Exit pupils of 8x20 binoculars (top) and 8x56 binoculars (bottom). As you can see, the 8x56 let's in a lot more light.
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My favorite configuration for hunting in the timber is 7x42. Compared to an 8x42, a 7x42 will typically have a wider field (easier to get on target fast), better depth of field (less focusing required), less hand-shake (easier to hold steady, especially one-handed), and a brighter image in low-light thanks to the bigger exit pupil. The only problem is, there aren't many 7x42 binoculars being made anymore. I have the Leica UVHD, but brand new they would cost nearly 10 times what you're wanting to spend. There's the opticron discovery which is in your price range and gets good reviews. The discontinued 7x42 meopta meostar is a great value, but unfortunately they don't come up for sale too often. A pair did sell on here not too long ago for $500. They didn't last long though, I tried to buy them the day they were posted and someone beat me to it.