A mechanically elevated rest like the Rock JR you mentioned elsewhere. The Bald Eagle and Sinclair rests are fantastic for the money, with less lash than the Caldwells, and greatly improved windage control. The Rock JR you had was about as basic as a machine rest can get, and the bag that comes on it, plus the inability to compress the bag with the top, just doesn't really support the rifle for tracking the way you want.
Here's one of the Sinclair models, I have - I have a couple of Protektor bags to mount to this rest which lock onto my rifle forends, but pictured here is an Edgewood rest bag (fairly certain).
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For a rounded sporter forend, stick one of these cheap EGW forend bag riders out front and party on.
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You get much, much improved tracking and better rifle stability by squeezing a square forend like this bag rider block between the sides of a bag like the one pictured above. Comparing this to the Rock BR, you can see how the forend would be better supported and can ride the front bag like it's on rails - truly tracking straight into and out of the target.
These will be a dramatic improvement to your forward end support to stay true on target from the bench.
For the tail end, a bunny ear bag is almost impossible to beat for tracking on target. This is a great, inexpensive edgewood model. I prefer leather tops, then I powder and burnish the heck out of them to let the riflestock slide between the ears.
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Alternatively - you can take an F/TR or ELR mentality to tracking, and use something like a Sinclair Skipod on a carpeted door mat under the front, with either an Edgewood or Protektor bunny ear like pictured above or something like a Cheney Airfoil rear bag:
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These are combinations of gear which track straight into and out of targets. The Sinclair skipod on a door mat with an airfoil bag is a very affordable set up.