One reason more is perfectly good English.
Why not "I have a reason more." ???
You must use the number (one) for this construction rather than the indefinite article (a). Equally, you could not say 'a more reason', as it makes no sense.
Could "di più" also work?
I have just seen, from another learner here, that "in piu" is used for nouns and "di piu" for verbs.
Ragione is also used as Right. quando usiamo per giusta o ragione?
Why not: i have a better reason?
That would be ho una ragione migliore. As an adjective, piu' means more or several.
Ho un'altra ragione....rolls off the tongue better.