The 30x444 (aka 309 JDJ) would also make a nice conversion. Would take a bit of research before cutting, as the 309 JDJ is a propriety chambering and JD won't rechamber your Handi. He's funny that way. So, what is available are close copies that are also based on the 444 case. The thing to be sure of is that whoever you get the reamer from will stand by it being true to the dies you get.
The 30x444 would give you a capacity advantage over the 30-40, the case is larger in diameter and the cae head to shoulder length should be close to that of the 8x57 Mauser.... 1.8" give or take a few thousandths. For a wroking idea of velocities, add 10% to what a 30-30 does and 15% over a 30-30 for heavy weights. 2700 fps for 150's and 200 grainers at 2300 fps should be realistic at modest 40,000 - 45,000 psi