Depending on your age you may need to qualify for the work test, they may have done away with it not sure.
Personal opinion is - if you do the concessional carry forward contributions then it will fill up your current bucket (27500) for this year 1st, the it will got to the furthest year 5 year prior, for yourself that will be the 2019/20 financial year.
Compulsory super is included in the total.
It may work for you to do a spousal contirbutiin between you and your partner.
You can only make these carry forward contributions if under $500,000 balance individual.