Here are some simulations that show the DC transfer characteristics of your original circuit as well as some changes you can make to improve it.
Notice that the output voltage of the first circuit starts at 1.25V and reaches the maximum output voltage at about a half turn of the pot--50% of the pot isn't being utilized.
The second circuit introduces R2, which sets a minimum output voltage. In this case, around 2.7V. Most blue LED's Vf is about 3.2V. The issue with the output reaching the maximum too quickly still hasn't been fixed.
The third circuit changes the pot's value to 1k. The resistor in parallel effectively helps make it a little smaller than that, which helps stretch useful range of the pot a little. The output voltage reaches its maximum at 94%, instead of 86%.