For the vast majority of hunting tasks in the United States, it's hard to beat the standard soft-tipped bullet put out by most of the major manufacturers. Remington Core-Lokt bullets are wonderful game-getters and they're inexpensive. Many of us, after trying/experimenting/using more expensive bullets are returning to the CL bullet because they are also wonderfully accurate. Sierra GameKings are wonderfully accurate, as are the Hornady line of standard bullets. I've got one load, in .243 Winchester, that uses Hornady's standard bullet that shoots into 0.4" at 100 yards. It's hard to argue with accuracy like that. I've seen Core-lokt loads that shoot into MOA.
I've also seen guys who ground off the tips of FMJ and left a jacket in the barrel. One barrel we sectioned back in the 60s had a jacket permanently swaged into the rifling. The guy shot one round, squirted the core, leaving the jacket in the barrel. Subsequent rounds went through the jacket, ironing it into the rifling. Accuracy degraded, predictably. He ordered a new barrel and when the barrel was changed we cut the barrel lengthwise to look at the rifling and throat. Ah-ha! There's your problem, right there.