Export segments with more fields, view contacts with more field

I want to be able to export segments with additional fields, such as phone number, address, etc. I cannot find any way to do this! When exporting a segment, fields seem to be limited to name, email, date added. I can export my entire list with all fields, is there any way to export my segment with all, or a selection of, fields? 

I would also like to be able to select any of my fields in "Modify Columns" to display in the main Contact view. 

Any ideas?