.243 Winchester in a Ruger American - quite affordable, and you won't be let down.
Or...
I'd weigh the medium and big bore options:
.35 Whelen
.375 H&H / .375 Ruger
.405 Winchester
.444 Marlin (if you reload - I hope so, if you shoot the Hornet much)
.45-70 (modern rifle - not something limited to low pressure)
.500 S&W
If you're really feeling froggy (and reload), you could investigate some wildcats or less common cartridges for a custom rifle...
.240 Weatherby
6.5x57mm
7mm STW
.411 Hawk
.416 Ruger
.50 Weatherby
...the sky is the limit.
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Don't even try it. It's even worse than the internet would lead you to believe.
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