You should do them at the same time. Use a starter fertilizer. You
can do it now, but the best time for seeding is in the Fall. If you
have any reasonable size, I'd recommend renting a slit seeder. They
generally go for about $60 for 4 hours, will save you a lot of work
and get the seed into the ground where it will have a good germination
rate. Some people buy $75 worth of seed and just sprinkle it
around, where little of it actually grows. Also, make sure you use a
good and proper seed.
now that we know your location we also know you are not subject to
intense hot summer sun as areas to the south of you are. it is that
hot intense heat that kills most seed just as it begins to germinate
and that's why fall seedings have produced better yields.
I second the recommendation you rent a slit seeder because getting
the seed into the ground provides a dramatic improvement in the
germination rate. The cost of the seeder can be recovered in the
area of improved seed germination. An accompanying lite application
of a starter fertilizer during an over seeding of an existing lawn will
improve the growth rate of new grass. However, consideration for the
grass type and its nitrogen requirements needs to be taken into account
before adding anymore nitrogen to the existing lawn.
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