Can modal verbs in the perfect tense, i.e. may/might/could have done, refer to the future?

Solution 1:

He may have arrived yesterday -perfectly alright.

He may have arrived now -just about scrapes in as grammatical but more usually one would say He may have arrived BY now.

He may have arrived tomorrow - IS NONSENSE. He may arrive tomorrow is the correct form of the future in this instance. In other words, with the modal 'may' or 'might', you do not need to use 'will' to indicate the future.

You could, of course, also say Maybe he will arrive tomorrow, which means the same thing.

Solution 2:

This 'past in the future' using the 'future perfect' {Englishpage.com} is certainly acceptable {A Communicative Grammar of English: Geoffrey Leech, Jan Svartvik: p 154}:

They will have arrived by tomorrow.

Using more central modals instead of the basic future-indicating will is not always acceptable:

Mary should have arrived by next week.

*/?Mary must have arrived by next week.

?Mary must have arrived by next week to stand a chance of seeing her grandchildren.

(?)Mary may have arrived by next week.

Mary should have arrived by next week.

*Mary may have arrived tomorrow (as WS2 says).

*/?Mary could have arrived tomorrow.

Mary could have arrived tomorrow, had she managed to get that flight.

[these examples my own, not Leech et al's]