When the holes are dug, and posts are placed in the holes, I dump in 3-5 gallons of water and kick the dirt back in around the post with my boots, using a re rod or shovel handle pushed up/down like a plunger in the hole to distribute the mud evenly.
The slurry of mud self levels with no need to tamp, and wiggling the post around also helps to get a full hole of backfilled dirt..
The next day the post are firm, just as if you used concrete.
I will admit Blue clay soil back fill is best substituted with a loam or other as the blue clay is lumpy.