OK - .30-06 = 7,62x63mm
Pretty much the "niche" inbetween that and the 7,62x39 is the .308 = 7,62x51....
Kidding.

I think that with that in mind, the .243 is the wrong choice. It is good for long distange and light game (fox, deer) but since the bullet diameter is small and the bullet is light, it is not between the .30-06 and 7,62x39.
A more suitable option the would be the 7x57mm or 6,5x55. The 6,5x55 excels also on long shooting distances.
There is good ammunition available for both, and they have taken their fair share of fairly large game, too. I would use neither for big boars at close range, however!
But still I don't think they beat the .30-06 with your criteria!
Fox and deer - the .243 is OK. Maybe wolf, too? But not for boar, it's too light. Given the option of those two I would choose the 7,62x39.
Accuracy and plinking - both your choices are good. For plinking the lower-recoil 7,62x39 with cheap ammunition available would be my choice.
Accuracy - depends more on the rifle than the caliber. I think no difference in real life.
Hunting - I have a hard time seeing the 7,62x39 as a hunting caliber much at all, but for your purposs I would prefer it over the .243 (which is an excellent cartridge!).

- Lars/Finland