Once you feel you've got it down at 100, move the metronome up a notch and play it at that slightly faster speed. On many electronic metronomes you can switch between single number increments (e.g. 100 to 101) and the more traditional gaps from the original pendulum metronomes. 1 bpm increase doesn't make that much difference to most people most of the time, so work the part you've learned well at the new speed until you feel comfortable and then increase the metronome again. Don't try to go from 100 to 120 in a single day -- take however long it takes. You can probably manage 2 or 3 increases every other day or maybe every three days. Following that procedure you can probably move from 100 to 120 in a week or two.
Work it up in sections, but don't leave one of the sections behind -- in other words, if you can do the first section at 100 but the other sections remain at 96, work those sections up one at a time until they're 100 also, so you can play the whole piece through consistently at 100. Then begin, section by section, to move from 100 to maybe 105, until you've got all the sections at 105, then begin the process all over to move to 110. Don't leave one section at 96 as you move another section to 120 -- keep the whole piece together so you can perform it for someone all the way through at a consistent tempo and with confidence.