Death-row inmate Hank Skinner was slowly chewing his way through the enormous cheeseburger that was supposed to be his last supper when good news arrived: he'd no longer be forced to digest the artery-clogging meal in an executioner's chair.

The US Supreme Court intervened late on Wednesday to prevent a lethal injection from being given to the convicted murderer, saying it wants more time to consider his appeal. "I'd made up my mind that I was going to die," the condemned man told guards. "I really feel like I won today."

Skinner has always maintained his innocence of the murder of his girlfriend Twila Busby and her two adult sons at their home in Texas on New Year's Eve 1993, and that the killer was the victim's uncle. To prove his version of events, he wants key crime-scene evidence, including semen samples and bloodied knives, to be DNA tested.

But several courts, of increasing seniority, have denied that request, saying that a convicted man has no right under US law to seek a review of evidence that his defence team did not get analysed at his original trial.

Even though the execution has been stayed, there is no guarantee that the Supreme Court will authorise tests that could either confirm or disprove Skinner's guilt. The justices merely want more time to study his appeal – and will probably not make a decision for several weeks.

In the meantime, Skinner – whose full menu for his last meal was two chicken thighs, a double bacon cheeseburger, fried catfish, onion rings, French fries, a salad with ranch dressing and a milkshake – remains on death row at the Walls unit in Huntsville, Texas, where his fate is the latest cause célèbre for campaigners seeking to abolish capital punishment. In recent days, the state's gung-ho Governor, Rick Perry, has been bombarded with letters from around the world asking him to delay the execution.

Skinner's supporters say he was asleep, having consumed drink and drugs when Busby was raped, strangled, and left on their living room floor with her head battered by an axe handle. He also slept through the murder of her sons, who were stabbed in the back as they slept, they claim.

At his trial, in 1995, Skinner's defence lawyer failed to have most of the evidence from the scene, including a rape kit, several biological samples, and the alleged murder weapons DNA tested. The jury convicted him after just two hours of debate.

Doubt began being cast on Skinner's guilt in 2000, when a group of students from the David Protess Medill Innocence Project at Northwestern University, who have been responsible for the exoneration of five others on death row, came to Texas to investigate his case.

They interviewed Andrea Reed, a star prosecution witness, who recanted her testimony that Skinner confessed. They also discovered that Busby's uncle, Robert Donnell, had a history of sexual violence and had made advances towards the victim that night.

It ought to be easy to establish if Donnell, rather than Skinner, fought with Busby before her murder: scientists would simply analyse flesh samples taken from beneath her fingernails, and blood found on the weapon.