Use insert edge loop, and select the created loop, go to translate mode, and then check preserve UVs in the tool settings, and move the loop as you see fit.
When i have a simple geometry its easy to produce viable UVs with automapping. The problem is when i use "insert edge loop" i cant make them at the grid lines. You can insert the edge loop and change them but this sucks because you stretch the UVs. Bevel is no option because its destroys your UVs much more. When you have a…