gnoahhh and flintlocke are on track! The Mannlicher-Schoenauer's were developed for exactly the usage the OP is taking about. There are many other suitable designs that work as well, but none better in my opinion!

flintlocke, I've had good luck accuracy wise with several bullets in my 1903 but now stick with the 160 gr Hornady RNs and 156 Prvi RNs because they both feed properly and others don't. I have also settled on JB's load of 40 gr of H4831 for both bullets. I wish I could still slip through the woods like I used to do, but at age 74 with very poor balance and bad knees I tend to spend 95% of my time looking at where I step instead of where the deer are. But every now and then the M-S still gets it done!

[Linked Image from i.imgur.com]