Would this scenario render someone effectively unable to vote legally?

[I spent the winter writing songs about getting better]

Let's say someone moves on November 1 and to a new state. As of November 1 they are no longer a resident of their old state, but probably don't meet the residency requirements in the new state as most require residency for a certain period of time that isn't long but more than a week (I believe Minnesota's is 21 days.) Is there any way to rectify it? Like they could vote early/absentee in their old state before they moved out but would this violate the old state's law?

Actually it doesn't even necessarily require a move to a different state. An in-state move could result in someone being unable to register before the registration deadline at their new address in time, but also no longer a legal resident at their old address and old precinct. Maybe states have laws that would protect such a person voting early/absentee at the old address but I don't know. In Minnesota and some other states this person could register same day if they had sufficient documentation of their new address but I don't know how it would work in states that don't have same-day registration.

[Ferguson97]

It would probably depend on the state, but I am fairly confident that they could vote absentee/mail-in for their old residence. They wouldn't be removed from the voter rolls on such short notice.

[Donerail]

States can't have residency requirements for voting for president; what you're referring to is a close of registration date. You can register to vote after that, you just won't show up on the voter rolls used for that election and will have to cast a provisional ballot.

[Del Tachi]

I haven't lived in Mississippi for 2+ years and I still vote there.  This isn't an issue. 

[satsuma]

I haven't lived in Mississippi for 2+ years and I still vote there.  This isn't an issue.  

I don't think it's voter fraud unless you're double-dipping. Simply for car registration and license reasons, though, I'd recommend making your relocation official. I procrastinated on that for a year after my interstate move and didn't encounter problems, but it easily could have been a worse inconvenience.

As to the OP, my state relocation was shortly before the 2016 presidential primaries, and upon researching that question, it definitely seemed harmless and legal to vote (in person) in my old state, when my residence in the new state wasn't yet official.