We consider the problem of minimizing the time to approach and land near a target radio beacon at an unknown location with an Unmanned Aerial Vehicle (UAV). We show that a cone-like region exists above the target inside of which bearing measurements of a directional antenna lose directionality: signal recordings in all directions yield similar signal strength. We present a geometric model of this region based on antenna simulations and data collected with a real system. Our main contribution is a strategy that takes advantage of a UAV's ability to change altitude and exploits a special structure occurring when approaching the target beacon from above to reduce the flight time required to land near the beacon. We analyze the performance of our strategy and demonstrate through simulations that by exploiting this structure we can achieve shorter flight times than our previous work.