If the convict has an otherwise clean record or perhaps nothing more than a misdemeanor or two, then by all means let's consider the new evidence and clear the man if there is any reasonable doubt.
However, if the rap sheet is a mile long (as they often are in these cases) with multiple felony convictions, history of violence and whatnot, then let the man rot in jail regardless.
I sat on a jury once for a murder trial and the prosecution was strictly forbidden from mentioning the past criminal history of the defendant. We (the jury) were even told to "disregard" the fact that he appeared in the courtroom in an orange jumpsuit.
After the trial was over, we were finally told during the sentencing phase that this guy was already serving a prison term for another crime (which was why he appeared in an orange jumpsuit) and had a long history of other crimes. Yet this was not allowed to be mentioned in court during the trial.
Fortunately we did convict the man but we would have deliberated a lot less had we been made aware of the prior violent and criminal history of this man. Some say each trial needs to be conducted in a "vacuum" of presumed innocence which included masking previous convictions. I do not agree with that opinion. I think previous criminal history should be relevant during trial and vice versa - as a jury would usually cut a defendant a little more slack if there were no prior convictions and this was his first brush with the law.
Having a record is no guarantee that you are guilty of a crime. Otherwise, just dispense with a trial altogether.
If you go to trial and are convicted, then your past should influence the sentencing.