Did you even read what you are buying, or just saw GRASS and ON SALE $9.99.
Though it could be ideal in this case because more seed is needed or it will fill the gap with weeds. So depending on weather, I would add actual perennial seed to this lawn to begin establishing it.
Do some research on lawn renovation. Annual means it won’t survive winter. Start your renovation in the Spring. Not as good as Fall, but you get what you get. Good luck.
I’d wait simply because you won’t be able to mow it if you put down more seed. Once it’s well established and mowed 2-3 times, then you can overseed if you want and the weather permits.
If the intent is to establish a ryegrass lawn then I totally agree with the previous posters that annual rye will usually die in the summer heat. Perennial rye is a much better choice.
I happen to live near a place where they sell sod for $1.78 per piece. It’s centipede grass so the look isn’t as rich as other grasses, but the price is awesome! What’s your zone? Have you done a soil test yet?
Annual rye is tough to create me by and experience are you sure? 1/3 of a 25 lb bag over 1100 sqft is a bit late light. I’d put down another 4-5 lbs after the 1st cut.
Since you’ve got some left in the bag, dump it on the thin spots.
let it ride then seed a little more