You might look at exercises designed to strengthen core muscles and particularly, lower back muscles. Doing that will speed up the process but simply spending more time on the drops will be critical. Stretching is also critical as much of the issue is finding better stretch, often in the upper back/neck. Your bike position becomes even more critical as your center of gravity is now further forward putting more weight on your hands and requiring a stronger core and possibly a saddle position a bit further back. Saddles made for people that are spending a lot of time on the drops often have a bit more of a curve at the back whereas upright riders typically are happier with flatter saddles. I realize people don't go from all upright to all on the drops but these are important things to understand and consider as you begin to spend more time on the drops. In the early season a 40mi ride with significant time on the drops will leave me with a really sore back part way through the ride. Now, toward the end of our road season I don't feel my back regardless of how much time I put on the drops. It will come, even if you do nothing but spend time on the drops but stretching/exercising will allow it to come more quickly and effectively.