When it's moving away from the sun, the sail is being pushed "forward", which adds to it's velocity- this increases the apogee on the opposite side of the orbit. When it's moving towards the sun, the sail is being pushed "backwards", lowering it's current velocity and reducing the perigee. As the perigee comes lower, it spends less time in that end of the orbit, where it's gaining velocity, and when it's apogee is high it spends more time in that end of the orbit, where it's losing velocity. It's not a winning strategy I'm afraid.
But imagine! Imagine if we had a slightly different solar sail. One that could retract and expand at will, via solar powered electric motors. Then we could have it open it's sails when the "wind" is favorable, then close them when it's not. It could slowly but surely escape earth's gravity well, and gracefully travel around the solar system. A big enough one might even be able to accelerate right out of the solar system.