Any additional drilling only serves to alter the price of oil due to speculation. No significant change in oil supply can be made in under a decade.
It's ok to drill if you want to worry the speculators to urge the price down, but the long term strategic energy goals should be toward other natural sources.
Ultimately there is a limit to how high oil prices can go because there is an economic ceiling created by the present cost of renewable energy sources (wind, solar, geothermal, etc). On a large scale, oil will always cost less than those sources (because nobody would use oil if clean energy sources worked as well and cost the same), so as the cost of delivering renewable energy goes down (due to manufacturing improvements and increased infrastructure), the price of oil will go down naturally.
Eventually we'll stop burning oil and just use it to lubricate things and to make plastic.