You may want to look for a Lee Enfield. They are accurate and you will never work a faster bolt. My No4 variant from 1943 came with excellent adjustable sights. The .303 round is also great for deer.
This was the basic rifle of the British empire from before WWI through the Korean war, so countless examples are available and you might be able to find one for a good price. Think of it as the British version of the US Model 1903.