Admittedly have not dealt with this in Greece, but if other countries are similar, the free
days are (much) more crowded than normal.
Assuming you are on a vacation, the cost of admission, especially in Greece, is not very
large compared to the overall cost of your trip. And navigating larger than normal crowds
to save a few dollars may decrease the enjoyment of your experience.
In any situation, going very early or very late will give you a better chance of having less
crowds, but it's somewhat relative based on how many people will go during that particular
day.