In documents filed Tuesday, the death row inmate said his previous attorneys didn't do a good job and the Mississippi courts have refused to give him a hearing and an expert to prove his "intellectual disability."
The Supreme Court had not ruled on his motion Tuesday.
Mitchell, now 61, had been out of prison on parole for less than a year for a 1975 murder when he was charged with raping and killing 38-year-old Patty Milliken.
Milliken disappeared on Nov. 21, 1995, after walking out of the Majik Mart convenience store where she worked in Biloxi to have a cigarette with Mitchell. Her body was found the next day under a bridge. She had been "strangled, beaten, sexually assaulted, and repeatedly run over by a vehicle," according to court records.
Mitchell was convicted of capital murder in Harrison County in 1998.
Mitchell argues the Mississippi Supreme Court twice refused to consider his ineffective counsel claims stemming from actions by his lawyers during the penalty phase of his trial and during his post-conviction petitions.
He said at no time did his attorneys try to develop evidence of his "intellectual disability" when evidence was available or could be available if he was given a psychological evaluation.
In a post-conviction petition, an inmate argues he has found new evidence — or a possible constitutional issue — that could persuade a court to order a new trial.
Similar arguments from Mitchell were turned down in the federal courts last year.