I was trying a new load with my .30-06 Vanguard, off of the bench. I also was using a Hoppe's rabbit ear rear rest bag for the first time, compared to just resting the butt on a sandbag. I didn't hold the rear butt like I used to, because of the Hoppe's bag, but rather the forestock. Most of my groups developed a lateral spread of 1-1/2", compared to a normal group size of sub-1". I wonder if there is a different technique I could use...grab the whole rear bag?

Make sure you are not "titting" the rear bag.
That is leaning against the bag.

You may be leaning your face hard against the butt stock. Try putting your left hand under the fore end for a few shots. When you lean down too hard with your cheek your left hand will feel it.