This deal from 2019 is back for 2020. Publix is offering a $10 gift card to get your flu shot and most insurance companies(iincluding medicare) will fully cover the shot with no out of pocket expenses under their "wellness" benefits. No appointment necessary and should be in stock at all the stores by now.
In the past Target has offered a $5 gift card and Walgreens a $5 off $15 "coupon" so Publix is still the best deal if you have one in your area.
If you use the Square Cash App you can stack their "10% off one purchase at any grocery store" (currently available) boost when checking out (of a food purchase, not the flu shot obviously as you'd have to have that first to get the gift card) along with the gift card to save even more money.
Most doctor's recommend getting your flu shot no later than November to cover the full flu season. It typically takes 2 weeks for you to build up a suitable amount of antibodies to fight off the flu.