What would increasing the accuracy do?
An edge is an edge. Your options are to decrease element size or change to the curvature mesher.
What you describe sounds like a bad model for analysis so anything but fixing the model is working around it.
Even if you could increase the accuracy in some way, if the features are all valid, the edges would still exist and need to be meshed.
put that way it is quite obvious, really.
Decreasing element size would lead to really tiny elements and the mesh is already curvature based...
Almost done rebuilding.