There are a lot of possible ways to solve your problem.
First, let's get you off the wrong track: this isn't a simple, line following problem. For it to be a line following robot, you need to lay track lines under your lawn for every place that you want to mow. Imagine a grid like a potato masher instead of the one your drew.
The problem seems to be:
Given a perimeter wire & an autonomous robot, how can I make a robot mow my lawn, and how can it do the job in a relatively efficient manner? (ie: don't take a random walk).
Here's what I would do:
1) Add a start/stop location to you map. You could use a strong IR emitter and sensor (test it to make sure your robot can see it in bright sunlight). Add this to your "charging station" and you've killed 2 birds with 1 stone.
2) Ensure that you're using encoded motors for your wheel motors. I'd take an approach similar to SLAM, but without the on-the-fly mapping. You can encode your yard as a map, making the code and problem drastically easier.
MIT's "SLAM for Dummies"
Also look into "particle filter localization".
3) Assume your yard is forever static. If it's not (if you plant a tree, remove the pool, or get a pile of mud after a storm, etc), we'll get to that in code with obstacle avoidance & with localization. But for now, we'll start with the simplest case, and we'll build up in difficulty from there.
Now with these 3 additions, I'd start the robot by hugging to the perimeter line -- this is its one little bit of line following. Let's start at the home, ie: the "charging station," and we'll work up the map towards the house. The root will straddle the perimeter wire. When the left sensor drops from 2V to zero, you're at the top of the map. Turn right, follow the line for x-inches or x-centimeters, turn right again, and go down the line blindly. This should work well until you reach the house's left perimeter line. You'll need to try this out in the yard and see how reliable your encoders are, how little slippage your gears give, etc.
Particle filtering will help your robot as it moves blindly to the next perimeter line.
The next issue to tackle is how to get around the pool. The encoded wheels will give you a decent idea of your positioning, so you can simply revert back to line following when you hit the pool perimeter. Follow the line for x-inches again, and turn. Unlike your exterior perimeter, your turns here are not 180 degrees. You'll need to experiment.
In lieu of trying to work that out, you can just avoid the pool area. Mow the easy part of the grid (everywhere that allows for simple 90 and 180 degree turns), and then come back to the pool at the end, where you're more likely to have poor localization.
I'd also add a sonar sensor at the front for obstacle avoidance.
I hope this spurs some ideas for you.