I assume you are thinking about the Remington 7600/760 or its predecessor, the Model 76. For hunting it is a great rifle. They are fast and accurate and legal in all states that allow rifles for deer hunting.
If you hunt small game with a pump shotgun, it should fit like a glove.
The only negative I can see with the rifle is that it will not be quite as accurate as the bolt action rifle typically. But they are plenty accurate for deer or bear hunting.
My Dad has one and I grew up seeing that as the classic "deer rifle". He has shot many a buck with that rifle.
If you are thinking pump "assault rifle", large magazine capacity are not available beyond 10; most are 5's which is what everyone uses for hunting. Come to think of it, Remington does make a urban version of that rifle.