Since I don't like the rimless cartridges in the Handi, I've thought about the .303. I think the .30-40 Krag would be a better choice.
It is a better choise, for the shooter that wants a 30-40.
I'm a fan of the 303. I talked to my 'Smith, and he told me to get a 30-30 extractor, and he would do the rechamber. I just haven't fallen over a used 7.62 yet.
In fact, he mentioned I could have one rechambered to 7.65 Argentine (another caliber I like), and use the original extractor. (Maybe I need to fall over two. Anyone have one or two they want to trade off or sell?)
By the way, this is the 'Smith at Capital Sporting Good, Helena, Montana. He is the reputable 'Smith who did my Handi 30-30 to 30-40 AI several years ago. He is even highly recommended by John Barsness.
Another plus of these older, tapered, low pressure rounds, is that you do not the accuracy and case sticking problems common to the 308s. (I know it is not popular to admit such on this forum, but I gave up on two Ultra 308s due to these issues.)