What size dent do you want in your bank account, for just the rifle? Wood or other stock? Stainless or carbon steel barrel/action? Without getting too expensive, you can look into Savage, Ruger M77, Tikka T3, Browning A Bolt, Howa, Thompson, Weatherby etc (including your Remington 700 choice), if you want to stay below $1000. Then you can step up to Browning White Gold Medallion, Sako 85, Winchester M70 Featherweight, Kimber 84, CZ, Steyr, Weatherby, etc if you want to stay below $2000 in general. Then you step into Hill Country Rifles, Christensen Arms, ER Chaw, Blaser, Nosler and the other "custom" rifles for the sky is the limit. The above is certainly not all inclusive but it was a wide open question.
Personally, I stay in the low to middle range. I have a 700 CDL in 30-06 with stainless action and fluted stainless barrel. Great rifle that is plenty accurate with a Zeiss Conquest 3.5-10. Also have a Sako 85L in 300 Win Mag with Swarovski 3-18 Z5 for those longer than 300 yard shots, a 338 Win Mag Ruger M77 Hawkeye Sporter with Simmons AETEC for the big boys at semi long range, and a 416 Rem Mag Winchester M70 in case Sasquatch or an enraged elephant wanders up to my front door
. I'd certainly love a Hill Country Rifles or Nosler but that is out of the question for now.
So what to get, other than the venerable 700, is a pretty broad question. Everyone will have a favorite, or two, or three, or .................... But in 308 you will have unlimited choices. Look at rifles that are 1/3 to 1/2 of your total budget and spend the rest on the glass. FWIW, I like controlled round feed rifles (kinda obvious from my stash) but the 700 is a fine rifle without that feature. The Sako has the smoothest action I've felt in any rifle short of a full blown custom or very high end rifle. Bottom line is you have a fun search ahead of you.