You can unwrap the selected portion, like you have in your image while leaving all the rest of the UVs untouched.
Starting with the UV layout you have in your bottom picture, in the UV editor, press P to pin the UVs. That will keep them in that layout, but only for the UV unwrap operation. All the other operations will ignore the pins if you have the entire sphere selected.
In this picture I first unwrapped the middle part of the sphere using Follow Active Quads (the pinned rectangular UVs), then one at a time selected the top and bottom and unwrapped with Project From View.
Thus what you can do is when you UV wrap the sphere only select the pieces you want want unwrapped, leaving all the previously unwrapped sections deselected.