So whenever I decide to stop playing and close the game window or go to Options and then Logout my character while I'm in the Clockworks, the next time I log back in I'm always back in the Arcade.
Is there anyway to go back to the place where I logged out instead of back up to Haven?
No. You must complete the whole dungeon in one run. It typically takes under an hour, so it is advised you set aside at least that long to play.
Consider putting this in the suggestions forum. I can't immediately think of any major game-breaking reasons this should not be allowed. But it is not allowed at this time.