.270 Winchester and .243 Winchester.
The former in particular is popularly discounted by the “in” crowd / 6.5 people, however the .270 remains a modest recoiling, flat shooting game hammer that by its very recipe (light bullets) has resisted the market push to heavy monos. It’s retained it’s bang flop aspects better than most as a result, and is available in every good rifle, and on every hardware store shelf.
The .243 is just a 75% scale .270, and is to the .308 what the .270 is to the .30-06. Another sweet shooting, quick, readily available and disproportionately effective round that was doing the mild but flat shooting thing long before the fancy options came along.
Getting away from readily available, the .250 Savage as the OP mentions, the 257 Roberts, and .257 Weatherby are all inordinately effective and light recoiling. The .30-06 is considered old hat but yet arguably the best rounded non dangerous game round in existence.