Legal hypothetical - inadmissible evidence and lives in danger

Crazy hypothetical here: cops are hunting for a kidnapping victim. One of them has some reason to suspect Joe Smith, but has no legal basis for that suspicion. So he does something illegal… let’s say he breaks into Joe’s house with no warrant, and while there, he finds absolute proof that Joe is the kidnapper, along with fairly ironclad proof of where the victim is being held (say, a factory Joe owns). But it’s a location that would require more than just his resources to access.

At this point, as I understand it, because the cop broke into Joe’s house, all of this evidence would be inadmissible, and could not be used to convict Joe. But at the same time, the cop knows where the kidnap victim is. Now, in real life I suspect that the cops would just fudge the law and go rescue the kidnap victim. But let’s assume that the cop now tells all to his superior, who is an absolute ironclad stickler for following the letter of the law 100%.

Would it be possible to get a warrant for the location of the kidnap victim to go rescue him/her, even though the warrant, being based on illegally obtained evidence, would not be legal for purposes of gathering evidence to convict Joe? That is, is there a legal principle by which the life-or-death emergency would override the illegality of the evidence for purposes of saving a life, but not for purposes of conviction?

Getting a warrant would be difficult, but if the cop went to the location, and broke in there and rescued the kidnap victim, he probably wouldn’t get into any trouble.

If he needs resources, he could go to where he knows the victim is, and just call for backup. He could lie and say he heard screaming in the house, or he is being shot at. Other cops will come and assist.

In the end, he could be (and should be) fired, but that is a small price to pay for saving a life.

The evidence is inadmissible at trial because the cop failed to get a warrant, but I don’t think it matters for issuing a warrant. If the cops even need to get a warrant to rescue the kidnap victim then I’m pretty sure a judge would issue it, and there are a lot of scenarios where they don’t need a warrant anyway. The problem is that any evidence obtained as a result of the initial illegal search would be inadmissible, and since going to the factory is a result of the illegal search, anything they find there (including the victim) can’t be used at trial. So the ‘heroic’ cop in your scenario probably just let the kidnapper get away with his crime.

Assuming kidnap victim survives and can ID, their testimony is still going to be perfectly admissible at trial.

I could almost see the defense arguing that it wouldn’t be admissible, simply because the victim wouldn’t have been found without the improper action.

I would hope that “fruit of the poisoned tree” wouldn’t extend to live testimony versus dead evidence, but without a professional readout I won’t assume it.