Police say Williams stole Corporal Cole's fully marked cruiser in Skowhegan and then drove to a Cumberland Farms on Waterville Road in Norridgewock, which he then robbed around 1:45 a.m.
The FBI told WGME they are sending assistance to help the Somerset County Sheriff's Office and Maine State Police with the investigation.
"The suspect should be considered armed and unsafe", a statement released by the Department of Public Safety said Wednesday morning.
State police on Wednesday were also staking out a house in Madison, Maine believed to be the home where the alleged killer lived with his girlfriend.
The cruiser was found abandoned just after 5 a.m. off Martin Street Road, Lancaster said.
"This is a very sad day for the Somerset County Sheriff's office, the citizens of this county, and people of the state of Maine", Lancaster said.
Williams had been scheduled to be at Haverhill District Court in MA on Wednesday morning for a probable cause hearing related to his arrest in that state March 22 on suspicion of having a gun without a license, said Carrie Kimball-Monahan, spokeswoman for the Essex district attorney's office.
Williams has a criminal record. State police there issued a nationwide alert on the search. In that event in late 2016, Cole took advantage of a moment when state police negotiators convinced an armed man in the store's parking lot to put down his gun. Cole was murdered in Norridgewock, Maine between 1 and 2 AM according to police.
"When we go to work, we put on a bullet-proof vest, and a side-arm".
Cole was described by a relative as a lover of music who was well-known throughout the community for his generous, peaceful and caring personality. "He was a good man, good family man, didn't bother nobody, didn't hurt nobody".
Gov. Paul LePage offered his support to Maine's law enforcement community as the manhunt continued through the state's central region. Eugene Cole of the Somerset County Sheriff's Department. The last ME law enforcement officer to be shot and killed was Giles Landry in March of 1989.
We'll continue to update this story as more information becomes available.