Practice, practice, practice.
Build your muscle memory so that anytime a monster goes offscreen, you either: quickly tap the DPAD (physical or panel) to compensate (because a monster that just went offscreen shouldn't be that far behind you to require more than two taps to get back into view), or quickly tap the Target panel (which you really should have on your touchscreen if you're having difficulty tracking offscreen monsters) and then tap L. After a few hours it should be second nature and pretty instinctive to do, which would make your reaction time that much faster.
By the time you get used to this, you'll be a much better hunter than players who use peripheral crutches (i.e., Circle Pad Pro, or Cstick users), because by then you are not crippled by not having access to those extra controls.