I have never owned a Handi Hornet, so I can't really speak from experiance about it. But I do own and shoot lots of 223's and have been experimenting with different loads for it using BlueDot pistol powder, and I am very happy with what I am finding. On the light loads it can mimic a 22LR, 22WMR, and the medium loads 22 Hornet, and then you still have the case capacity to use full power high velocity 223 loads to reach out there.
The light loads, around 22LR and 22WMR speeds, are pretty quiet and very easy on the ears. And for safety sake, I never use anything hotter without ear protection.
The 223 is a very versitle round and a ton of fun to play with, as I am sure the 22-250 and 220Swift are also-in which I plan on diving into one day

. The 204 is great too, but the bullet selection is smaller then the .224's so it's not quite versitle.
It's just an idea to throw around, in case you are a reloader.
I do see how the 22Hornet has a lot of character, and has a lot of history to it. So I ain't knocking the little guy by any means. Many here love theirs too much and I have seen too many with good results to speak down of it.
I just find the 223 to fill almost all my personal varmint needs, and that includes Rabbits and Squirrels.
-Aaron