I have to wonder why you want to use something else when your 243 and load has produced such great results. If it ain't broke why fix it just comes to mind!
The 06 with 150-165 gr bullet will work well on deer. Like mentioned if you want to try then do some research and pick up a few brands/styles to try. Just let the gun tell you which is the best combo to use. Personally I like Ballistic Tips or Boattail bullets, these bullets usually offer great accuracy and work well on deer sized game. All major manufacturer's have ballistic tables and info on their websites.
I agree if the front shoulders were both broke the deer would make it to far, impossible to run with no front wheels
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