On an established lawn, I'd wait until late summer/early fall which is
the optimum time. You have cooler temps, and twice as long for the
grass to get established before getting into the summer hot months.
Because of cooler temps, less water is reqd and its easier to keep it
wet while it germinates. You also have a lot less competition from
weeds to deal with. Early spring is the second best time, and the
earlier the better.

Either time you do it, IMO, just throwing seed around is a waste.
Given what good seed costs, it's more cost effective to rent a slice
seeder, which will cut grooves, put the seed in contact with the soil.
You will get much better germination, which is what you want.
Spreading seed around, you're likely gonna keep doing it several times,
spending more money and a lot more labor.