Don't rush out and get a 3-9X40 scope, not if you plan on shooting beyond 300 yards. The reticle is too heavy, and covers too much of the target at longer distances. Also stay away from those little circles, and post reticles as well. Get a scope with a fine reticle, the number of lines are not important as long as they are fine lines. People will tell you that you can not see those lines to make a fast shot, so what. You are not making fast shots at long distances. Plus with practice you will still be able to pick up those lines fast at close ranges.
In most cases the game does not know you are anywhere around. They are calmly feeding and moving very little. You have all the time in the world so don't rush it. If you do you will miss. Do everything you do on the bench, just like you do on the bench. Consistancy that is the critical most important thing you can do. You need to have good clear glass to see any grass or branches between you and your target. I've missed many a Deer because of a small twig.
Each cartridge has it's own advantages and disadvantages, those have to be taken into consideration when shooting at longer ranges. A cartridge that shoots great at 100 yards and has good killing ability, may not work out at 400 yards. Yet an old cartridge that people laugh at for shooting beyond 100 yards can be very efficient at killing game at 800 yards. Case in point the old 45-70. 400gr Jacketed bullet, Muzzle velocity 1900 fps, energy 3200 ft pounds. 800 yards 983 fps with 858 ft pounds of energy. Trajectory like a rainbow, and bullet speed so slow you think it will never get there. But when it hits a deer at 800 yards a big hole in and a big hole out. The Deer usually stays put, if it does run it don't go far. Make the same shot with a .243, you will probably lose that Deer. Velocity is a lot higher, but the energy just is not there. So learn the limitations of your cartridge.
For extreme long range I shoot a .338/378 Weatherby with a 225gr Nosler Accubond traveling at 3210fps. At 1000 yards I still have 2174 fps left and 2361 ft pounds of energy. Your .308 shooting a 165gr bullet at 2910 fps muzzle velocity will have 3102 ft pounds of energy at the muzzle. At 300 yards your velocity will be 2232 fps and energy will be down to 1825 ft pounds. At 700 yards you will be down to 1463 fps and 784 ft pounds of energy. So again know the limitations of your cartridge.
Yes the Handi will hit them out to 700 yards, but will the cartridge do the job of killing the target at that range.