Moose Masher, just a thought, but you might want to clarify what you mean by "classic" cartridges. I think many people here assume that means "different", hence all the Euro suggestions, but I'm not 100% sure you meant it that way.
I didn't realize originally that the rifle in question was a single shot and very lightweight, in that case (and given you already have a nice 9.3x62) I agree that going with rimmed cartridges and going a step down in size may be of benefit. Maybe something like the 303 Brit combined with the 5.6x52R (22 high power) would fit the bill? Or the 22 hornet as I think you mentioned before, more available on this continent probably but less powerful than the high power.
I think that if you want "classic" in a single shot, you should seriously consider the 303 Brit as it is the cartridge that MADE Canada and up until the last generation, it was what everyone had. It has alot to say anywhere across the British Empire from Africa, to India, to Canada. It walks that fine line between still being available, but just being unique enough to be classic, and it can do anything the 308win or 7x57 can do. I'm thinking in your case starting with the 303 brit and then deciding if you want to go up or down from there would be a good way to start. If you want bigger, the 45-70 would make a great, classic, and flexible cartridge that can take anything the 9.3x62 can with heavier handloads, or you can load it down or use the soft shooting factory loads. Another choice would be the 9.3x72R as others already mentioned. If you want to go down, the 22 high power or 22 hornet would both be options. Which way would you want to go?