Thanks for the explanation. Have a look at another example, please.

She should have phoned her mother to warn her that she might fail to come.

Can I replace "might" with "may" here?  The speech is about the events taken place in the past and the second part of the sentence should be used according to the rules of consistency of tenses. As far as I I know, the verb form "might" is some kind of past form of the verb "may", therefore "may" cannot be used in this sentence. 

Correct me if I am wrong.