I saw a TV program called "Ask This Old House". One of the guy in the
show showed the following way to distribute top soil (or compost?)
through out the lawn:

- Use a snow shovel that is deep (not wide).

- Put soil/compost on it.

- If you are right-handed, put the handle of the shovel under your
left arm-pit, support the shovel with your left hand, and use your
right hand to sweep/broadcast the soil/compost from the shovel onto
the lawn.

- You will also need to use a rake to smoothen area that has too
much soil/compost in one place.

I tried just that yesterday in my small front yard. This worked. But I
was so tired after that (it was hot and humid yesterday); I cannot
even bring myself up to do jogging in this morning. I will do this
again in fall when I need to re-seed my lawn; at that time, I will use
an aerator as suggested in another message in this thread.

Hope this helps.

Jay Chan