Hayes - aka Grant Haze - is charged - along with his current wife Amanda - with killing Ackerson at their Raleigh apartment in July 2011 during a long-running custody dispute.
Prosecutors say they chopped up Ackerson's body, put it in coolers, and then drove them to Amanda Hayes' sister's house in Richmond, Texas, where the body parts were dumped in a nearby creek.
In court Tuesday, prosecutors set up testimony with evidence introduced Monday, which included trash taken from a dumpster at the apartment complex where Grant Hayes and his wife were living at the time.
Authorities found Ackerson's DNA on a latex glove, along with scrub brushes, bath mats, a plunger, masks to help deal with odor, ice bags, blood stained clothing, and bleach soaked towels.
The latex gloves pulled from the garbage provided the first physical evidence that Ackerson's body may have been in the Hayes' apartment.
A SBI DNA analyst told jurors that much of the DNA evidence from the apartment may have been destroyed by cleaning agents like bleach.
Later in the day, jurors got to see and hear emails between Hayes and Ackerson in which the two bickered over their custody battle in the months and days before she was murdered.
Ackerson was the mother of Grant Hayes' two oldest children and she had lost custody of them to Hayes and Amanda, who also had a newborn baby with Hayes. Ackerson was fighting in court to get her kids back when she was killed.
Hayes' defense maintains he didn't kill Ackerson, just helped dispose of her body. It says Amanda Hayes killed her in a fight and that Grant is only guilty of trying to cover up the death.
Hayes and his wife are being tried separately. A jury of eight women and four men is hearing the case. Attorneys have also chosen four alternates. Prosecutors are not seeking the death penalty. Hayes faces life in prison if convicted. Amanda Hayes is expected to go on trial next year.