Yeah, really too late now--you want it in and growing before the "regular" grass is dormant, and before it goes to "dead."
Best time is in the fall, just before the first cold norther in october-novemeber. Better than that, is after a mild rain and before the norther. Hold off mowing for a week or so after the over-seeding.
After this drought, over-seed by about 100% (i.e., if seeding 1000sf, buy rye seed for 2000sf). That way all the hungry seed-eating birds will leave "some" behind (another reason to be out in the near-darkness before that norther, the storm/weather change will keep the birds roosting a day or two).
Now that you have dead grass, it's a pain. You are getting mud, so it's hard to get the seed to "stick" on the loose friable material when it is dry; and it all washes away in the gully-washer rains of winter.
This is where hydroseeding can sometimes be to your advantage. (Get references, ask for addresses; talk to previous customers, first.)