It is not quite clear to me what you are trying to do. But one solution may be to look at the signals on an oscilloscope. For example, here is a screen shot that I put some annotation on:

It is measuring the time for muon decay in the lower trace with a plastic scintillator. The oscilloscope also shows the signal from a NaI detector positioned above it. As you can see, the NaI detector is not as fast as the plastic scintillator.

(And the oscilloscope is again live on the web here, now with 5 liters of scintillator liquid).

It is not quite clear to me what you are trying to do. Why not just turn up the voltage of the PMT? But one solution may be to look at the signals on an oscilloscope. For example, here is a screen shot that I put some annotation on:

It is measuring the time for muon decay in the lower trace with a plastic scintillator. The oscilloscope also shows the signal from a NaI detector positioned above it. As you can see, the NaI detector is not as fast as the plastic scintillator.

(And the oscilloscope is again live on the web here, now with 5 liters of scintillator liquid).

Another thing to do is coincidence spectroscopy using the stereo sound-card input of a computer. Time resolution is low then (limited by the audio range). There are free software from different sources. A very simple way to hook things up is just a passive low-pass filter.