EPPING, N.D.—On a recent subzero day at a rail station here on the plains, a giant tank train stretches like a black belt across the horizon—as far as the eye can see. Soon it will be filled to the brim with light, sweet crude oil and headed to a refinery on Puget Sound. Another mile-long train will pull in right behind it, and another after that.

Increasingly, scenes like this are being played throughout the country. "Hot Trains" dedicated to high-priority customers like United Parcel Service Inc. roar across...