You should not need to go lower than a 150 grain bullet with a 30-06.. If you need less recoil, you need a smaller calibre. For example, a .284 in 140 grain will have better ballistics than a .308 with a 150 grain bullet. Same, a .277 calibre with a 130 grain bullet will perform about the same as a 140 grain .284 calibre bullets. The .264 calibre 125 grain bullet will perform about the same as a 130 grain .277 calibre bullets.
Then why not use a real small calibre bullet? Well, you have higher performance but with less weight; a bigger bullet at the same speed on impact will do more damage.
Even if you are only target shooting the lower ballistic coefficient will be less accurate, usually.
All gun folks have their own ideas, some backed by science, some by experience, some just believe what they believe. I would say, the 30-06 is best with a 165 grain bullet or heaver, while a .284 will do as well with a 140 grain bullet, yet the 165 grain bullet will have more effect when it hits the point of impact, at the same speed.
To make it simple, if you use a 280 Remington, with a 140 grain bullet of most game, it is going to be effective, same with using a 270 with a 130 grain bullet, yet the 280 will still be a bit more effective, as will the 270 with a 140 grain bullet, now all you need to know is how fast is the bullet moving when it reaches the point of impact? I would say the 270 with a 140 grain bullet will do about the same as a 280 with the same weight bullet, yet many will disagree, how many, as many as there are who prefers one calibre over the other.
For you, you’re looking at recoil, it seems to me the 30-06 is too much rifle for you, and a smaller calibre may be what you need.
There are other means to tame recoil, such as a recoil muzzle break, which is very loud, or as one man said, using a better recoil pad, this really does help. There is also a way to use a weight in the stock, which moves against the recoil, by making it slower, but such is often not used because it adds weight. Speaking of weight, the heavy rifle will tame kick more than a light rifle, yet if you need to carry it far, the heavy gun may be a determent.
I like the 260 Remington for most game I hunt at long distance; however, if I were to hunt an Elk, I would rather use a 30-06 with a 180 grain bullet.
I am sure more members here will fill in the many details I left out. What I am saying is if you do not get what someone said, my explanation may be informative.
caliber and calibre are the same word.