I wouldn't convert a nicely balanced stalking rifle in 9.3x57 to 9.3x62; you already have something very nice, so why change it? What game are you hunting that the conversion would be useful?
That out of the way, I took a plain-Jane H&R 330 (FN98) action and rebarreled (25.5 inch Lothar Walther) to 9.3x62. I also installed a Timney and a M70 style safety. I need to upgrade the stock (Hogue, currently) to something better, but I'm very satisfied with the project, on the whole. The rifle is no lightweight and is a pleasure to shoot, even with the full tilt (BigGame/286 gr bullets) loads.
I like the rifle very much, but I recently bought a Mannlicher .30/06 which is much more a "stalking rifle" design, if you catch my drift. Both rifles are great, but they are very different. My point would be that you shouldn't try to make the rifle into something that it's not, when it's pretty darn good as it is.
