You don't qualify because your eligibility to use the 2 year/5 year rule expired in 2016. If we assume you moved out on June 1, 2013 and turned the home into a rental, then your eligibility to qualify for an exclusion ended on May 31, 2016. Since you kept the home as a rental past that date, the exclusion rule no longer applies.
The point of the 24 month rule is that you must have owned the home and lived in it for at least 24 months to ever use the exclusion. If you move out after owning less than 24 months due to a hardship, you might qualify for a partial exclusion. For example, you buy a home on June 1, 2016, and move out June 1, 2017 for a hardship--less than 24 months. If you sell (instead of converting to a rental) you can take a partial exclusion.
Assuming you owned the home from at least 2011, then you already meet the 24 month rule and the hardship exception is irrelevant. What you are seeking relief from the is other part of the 2 year/5 year rule, that you must have lived in the home at least 2 of the 5 years ending at the date of the sale. The only exception to that part of the rule is if you are displaced from your home for military duty or foreign service. You could have sold any time between 2013 and 2016 and excluded your gain. By keeping the home as a rental longer than 3 years after moving out, you no longer qualify for the exclusion. Perhaps it would have been wiser to sell the home before your eligibility ran out, but you probably could not have predicted the layoff.
I don't know anything about "holding out for sale" versus actually selling the home...but I think you would have a tough case and would lose.
