Get a core aerator, the kind that pulls out plugs of soil. Spike aerators and those spiked shoes just push holes in and pack the soil tighter around them. Most tool hire places rent core machines, and splitting the day's hire with a neighbour is common.
Do it when the soil is moist, the day after rain or after a good watering, or the tines barely go in. Flag the sprinkler heads and any shallow cables first. Timing depends on the grass: early autumn or spring for cool season lawns, late spring into early summer for warm season ones.
Leave the plugs on the lawn, they break down in a week or two. Straight after is the best time to overseed and feed.
