.243 Winchester in a Ruger American - quite affordable, and you won't be let down.
I'd weigh the medium and big bore options:
.375 H&H / .375 Ruger
.444 Marlin (if you reload - I hope so, if you shoot the Hornet much)
.45-70 (modern rifle - not something limited to low pressure)
If you're really feeling froggy (and reload), you could investigate some wildcats or less common cartridges for a custom rifle...
...the sky is the limit.
"Such is the strange way that man works -- first he virtually destroys a species and then does everything in his power to restore it."