"may even have" or "may have even"?

Solution 1:

I believe this is a matter of adverb placement. Even is used here as an adverb and have is an auxiliary verb. The Chicago Manual of Style holds as follows:

"When an adverb qualifies a verb phrase, the normal place for the adverb is between the auxiliary verb and the principal verb {the administration has consistently repudiated this view} {the reports will soon generate controversy} {public opinion is sharply divided}. Some adverbs may follow the principal verb {you must go quietly} {are you asking rhetorically?}. There has never been a rule against placing an adverbial modifier between the auxiliary verb and the principal verb in a verb phrase. In fact, it’s typically preferable to put the adverb there {the heckler was abruptly expelled} {the bus had been seriously damaged in the crash}."

Based on this, I would say that the former sentence is more aligned with what the CMS recommends. This doesn't mean, however, that the latter sentence is incorrect.

This is also from the CMS; you might find this useful:

"The combination of an auxiliary verb with a principal verb is a verb phrase, such as could happen, must go, or will be leaving. When a verb phrase is modified by an adverb, the modifier typically goes directly after the first auxiliary verb, as in could certainly happen, must always go, and will soon be leaving. The idea that verb phrases should not be “split” in this way is quite mistaken..."