Agreed on letting the cores dry a bit. My perfect Spring yard treatment goes like this: 1) Core aerate. 2) Top dress with 1/4 inch of compost. 3) Light drag with chain link gate.
For optimum results, you need perfect conditions or some help (irrigation). The ground needs to be just the right moisture content for aerating. Neither too wet, nor too dry (irrigation helps a lot with this one). Then you need a couple of warm dry days to let the cores cook. Then the topdressing and dragging need a dry day. Then you'd like the skies to open up and give it a good soaking (or turn on the sprinklers for a double cycle).
If my yard was cool season grass (fescue, rye, etc.) I'd be thinking pretty seriously about getting this project lined up to happen within the next couple of weeks, here in Southern MO. I've got zoysia, so it will likely be mid-April or so before I'm ready. Either cool season or warm season grasses, you want them actively growing and out of dormancy before you aerate.