The Cuban Missile Crisis, as a whole, is always considered the closest "incident" for close calls, because it is not some kind of mishap but a framework in which any mishap or misjudgment could have easily led to nuclear use and escalation.

If one moves down from that, the "War Scare" of 1983 (Able Archer and the events that preceded it) is often considered the second-most likely, for similar reasons: it's not just a single mistake in a system or anything like that, but a large-scale set of circumstances that meant that misunderstandings could be amplified and not readily understood as misunderstandings.

Outside of those, I am not sure there is any kind of consensus on what would be the "next closest call." There are questions about how close the computer errors of the 1970s were (how likely they would escalate, since they came "out of context"), and there are some doubts (for the same sorts of reasons) about the Petrov incident (though the context of 1983 was inherently more dangerous). It is not entirely clear whether the Black Brant incident should be considered that close a call or not; some have said yes, some have said no. And there are a whole host of nuclear accidents that are not clear to have been "close calls" at all; it isn't clear that they threatened actual war, even if they threatened loss of life.

There are more creeping contexts, such as the Korean and Vietnam wars, both of which saw their advocates for the use of at least tactical nuclear weapons, that are not generally categorized as "close calls" but perhaps ought to be thought of in that way, given that it would not have taken too much change of heart from the Presidential leadership at the time to see them "go nuclear" to one degree or another.

But this isn't the sort of thing that historians get together and vote on. The above is just my sense from reading the various literatures around these incidents. Because we are dealing with pure counterfactuals (things that did not occur), it is easy to imagine that something is overblown or under-appreciated. But my answer about the Cuban Missile Crisis seems pretty unambiguous: pretty much everyone agrees, including the participants on both sides, that nuclear war was only avoided by a near shave.