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tl;dr: In the U.S., can a "not guilty by reason of insanity" (a.k.a., NGRI or NGROI) defense be offered after a guilty verdict but before sentencing?

In criminal prosecutions in the U.S. it is common and generally considered prudent for a defendant to refuse to expose himself to questioning at trial. This forces the prosecution to use independent evidence to prove beyond a reasonable doubt that the elements of the crime charged were satisfied.

A defendant may want to entertain a claim that he was insane at the time a crime was committed, but avoid entering that claim when there is a significant probability that the state will fail to obtain an conviction on the merits of the case against him.

So my question is: If a jury returns a guilty verdict for a crime, can a defendant subsequently (but before sentencing) argue to the court that he was insane? Or does that defense always have to be offered at trial for the crime, before the case is sent to the jury?

I assume that the person alleges insanity at the time of the crime, and not incompetence to stand trial. The best but dim hope would be to introduce evidence as a mitigating factor for sentencing – this would not avoid a conviction, but you might think it could reduce the penalty. Oregon v. Guzek, 546 U.S. 517 says that you cannot (automatically) introduce mitigating evidence after the trial:

the State may limit the innocence-related evidence he can introduce at
that proceeding to the evidence he introduced at his original trial

This case does not say that mitigating evidence can never be introduced at the sentencing phase, it says something complex about Oregon state laws on the matter, 8th and 14th Amendment rights, and 28 USC 1257(a).

Taking examples from Washington, you either plead guilty, not guilty, or not guilty by reason of insanity. The rules also state that

Written notice of an intent to rely on the insanity defense, and/or a
claim of present incompetency to stand trial, must be filed at the
time of arraignment or within 10 days thereafter, or at such later
time as the court may for good cause permit. All procedures
concerning the defense of insanity or the competence of the defendant
to stand trial are governed by RCW 10.77.

RCW 10.77.030 says

(1) Evidence of insanity is not admissible unless the defendant, at
the time of arraignment or within ten days thereafter or at such later
time as the court may for good cause permit, files a written notice of
his or her intent to rely on such a defense.

(2) Insanity is a defense which the defendant must establish by a
preponderance of the evidence.

This precludes making the argument after trial. (2) is a question of fact that has to be made by the jury, meaning that the evidence for and against the claim must have been introduced at trial (and there's no do-over if you don't like the outcome of the first trial).

No, the defence must be offered at trial

And, typically, the NGRI plea must be entered when the defendant pleas - courts do not generally allow a change of plea.

Your proposed strategy doesn’t suffer from the plea. NGRI is an affirmative defence - the state still needs to prove their case, if they don’t the correct verdict is not guilty. The judge will direct a jury to find on that before they consider if the defence has made their insanity defence.