If you are creating the segment from the Workspace then the segments will not give you the option of 'Save As' if those segments are not public. So, you can check the check-box of making the segment public while creating the segment from the Workspace.
If you want to create a new segment using the segment logic of the previous segment. Then I will suggest you to please open an existing segment & add the recent changes. After making the changes, please give the name to the segment and then click on the 'Save As' button as shown below.