Hello, Bob.
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I am not aware of a sure-fire ways to solve a looped-maze by simply modifying the left-hand rule. Did you have something in mind?
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The robot will likely do many turns in both directions while solving the maze; you could probably determine which direction the goal is facing by just keeping track of the number of turns, but you will need to do more to determine where that goal is (and how to get to it again). If your maze is grid-based like the one in our LVBots competition, keeping track of the robot’s relative heading is just one step further than counting turns.
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I am not familiar with any methods of solving a looped-maze without keeping track of the robot’s relative heading and some sense of where it has been in the maze. There are typically two phases of maze solving. The first phase involves learning the maze and usually not caring about how much time it takes; then the second phase is using that information to solve the course as quickly as possible. With some luck, you might be able to eventually solve the maze in the first phase without keeping track of the relative heading of the robot, but it would be necessary for gathering the kind of information you would need to solve the maze in the second phase (which is essentially a digital representation of the maze). More details about the method I used can be found in this post that you alluded to.
Brandon