I've been doing this for a few years now.
What I have learned that helps a lot for the garden not ending up completely mad:
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Mow at a higher height earlier in the year, but consistently.
This allows the smaller shoots of grass to establish.
Which in turn means that when you finally chop after May, you're left with relatively even grass.
Rather than chopping down fewer very long pieces of grass, and finding lots of under-grown gaps. -
Stop mowing the edges in April, and possibly pick one patch to leave un-mown until august. I normally do a corner.
The next year, you'd never know. -
Try to ID weeds that start growing, and hand-remove if required. Myself, I try to keep the Bindweed and Virginia Creeper in check.
I leave dandelions alone. There is another weed I've never got around to ID'ing, but I leave it alone now, and the bees love it. -
Keep them mower blades sharp. It helps prevent the grass getting ripped out of the ground when you do mow.