… how it all might have started
… how it might all have started
A fine point of meaning. I have also considered a close variant such as:
… how it might partly have started
May draw attention to the various ways in which the matter may have started. Partly is acting adverbially on start
… how it partly might have started
May draw attention to the role parts of it played in starting the matter. Partly is acting in an adjectival manner on it
But this is straining to find nuances that, while possible in this variant, do not seem to work with your own example. I therefore suggest that there is no significant difference there.