Compost after overseeding?

So, I aerated my yard then added fertilizer followed by Tall Fescue Seed.

I bought some Compost & Manure. Should I add a layer on top of the seed or is this not necessary? If not, when would be a good time to add the compost? I read that compost should be added before fertilizer and seed.

My second question is I probably should have gotten a Core Aerator, but I ended up using the Yard Butler Fork that just pokes holes in the ground. Should I ditch this tool and go with a Core Aerator...

Compost after overseeding?