I dont have steam myself but are you using steamworkshop for mods? If so does that have an option to auto update mods?
Is there a log you can check to see just what updates you are getting? Sounds to me more like some update is failing, so Steam keeps trying to do the update.
Steam wont let you start the game if there is an update for the game itself until its done as far as I am aware. Never actually heard of it updating
after playing a game, always before playing.