With no further specifics than "I want a rifle of .270 or greater caliber for under $1000," I'd say you're in an enviable position.
You can be free from lots of biases and preconceptions and hit the market to find a great deal!
I'd strongly suggest taking a couple of days to visit 5-10 local shops and scour their used racks. There were millions upon millions of fine rifles made throughout the last century by Remington, Winchester, Marlin, Ruger, Savage, and a host of others (not to mention a huge pile of Springfield, Mauser, and Enfield service rifles that were "Sporterized" with great care by skilled gunsmiths) and a large percentage of those rifles are now back on the market. In the great majority of cases there's absolutely nothing wrong with them, but the original owner has passed away, there's no-one in the family left who shoots or hunts, or the "newest/biggest/fastest" bug bit someone and they had to have this year's model and so sold several reliable old guns to finance the newest flash in the pan.
Or maybe it just wouldn't shoot less than 3" groups at 100 yds. So what? That's plenty more than adequate for 99% of the hunting shots taken each year. And chances are, the rifle that someone sold to buy into some company's "sub-M.O.A." guarantee would outshoot its owner any day of the week! A little experimentation with various ammo (especially if you learn to reload) will cut many rifles' groups in half.
I'd set a limit of about $400 and just browse with an open mind. You don't mention a preference for a specific action type. That's good! I'd bet a Marlin 336 lever gun would make you pretty happy. Or maybe a Remington 760/7600 pump rifle! I see them for right around $300 frequently and that's a fantastic rifle for such a price.
Any caliber over .270? That covers a lot of ground, but it's also kind of an arbitrary cut-off point. I'd argue that anything over .243 will probably get done all you need to do at this point, but you really aren't limiting yourself. Notwithstanding millions of pages of advertising text and gun-writers' words (same thing, more or less) there's little practical difference between most calibers in a given power range. Pick one, learn to shoot it well, learn its trajectory and limits of its power and you can take most game. (If you wanted to hunt Grizzly bears I'd assume you would have mentioned that!)
I'd be very surprised if you couldn't find some rifle to love, with a serviceable scope already on it and probably a sling, too, for under $350, without a whole lot of searching. If you spend the other $650 on ammo, targets, and range time, you and that rifle will be an impressive pair!
Good luck!
-Sam
