I have thought about this a lot as I mostly hunt the Blueys and just use my .308.
Use 2206H as it goes well in the short barrel. Any soft 150gr bullet would be ideal. Some have mentioned lighter weight bullets, personally I wouldn't use them.
I actually think @Marty Henry is on the money with soft round nose 150gr bullets for a .308 that is dedicated to close to moderate range shooting on fallow. I have a box of Speer 150gr RN sitting in the draw for this very purpose. I just don't use them as my .308 is my everything rifle, so I go with 165gr SP or SSTs, which of course kill fallow just fine.
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