By long range, I mean able to hit a man size target at 1,000 yards.
I am still thinking a Savage .308 Model 10.
Savage is a fine maker of rifles for well under a $1000... also, Marlin and Mossberg seem to make fine rifles at comparable prices.
More expensive but still under, or close to, $1000 is Weatherby, Remington, Tikka, Winchester and Ruger. These are excellent makers as well...
A varmint rifle in bolt action usually has a longer and heavier barrel which adds to accuracy, so that is something you might want to consider... nevertheless, in a non-magnum caliber, the 30-06 and .270 are your best cartridges in terms of accuracy and energy at 1000 yards. The .270 is VERY flat shooting, thus accurate, but the 30-06 in a little bit lighter grain than typical 30-06 rounds are very flat shooting, too, yet retain their power at long distances. These accurate, yet powerful, cartridges have some appeal due to their lack of recoil and kick relative to magnum rifles. Btw, the .308 is an EXCELLENT cartridge, too - both accurate and powerful, but it's not quite as accurate at such long distances as the .270 and 30-06.
In a magnum cartridges, the 300 Winchester magnum (or 300 Weatherby magnum) and the 7mm Remington magnum are EXCELLENT cartridges - both very accurate and even more powerful at longer distances. Btw, "Winchester magnum" and "Remington magnum" merely denotes the specific caliber designed and cartridge used, but you can get it, for example, in a Ruger or Savage.... such example is a Ruger 7mm Remington magnum or the Savage 300 Winchester Magnum. Getting a 7mm Mag does not mean you have to get a Remington rifle, rather, you can pretty much get it from any manufacturer.
The military (sniper teams) recently made the transition from .308 caliber to the 300 Winchester magnum, mainly due to the added maximum range of effective kill. The only downside to these magnums is the louder report ("BOOM") and the heavier kick (recoil).
You should be able to find ballistics charts. Winchester has a an excellent comparative chart but you have to download their software to use it.