Try adding one note at a time at the new tempo. For example, if you have four sets of sixteenths, play to the point where it breaks down. Then, add the next sixteenth at the new tempo. Once you get that down, add the next, then the next, etc. Over time, you train your hand/brain to get it together.
Sidenote: The importance of playing with a proper hand position cannot be overstated. In addition to allowing the fingers to move more fluidly (and in time), it also saves wear on the valves.