The vehicle was running but unoccupied when the officer pulled up, radioed in the license plate number and realized the car was stolen, Pugel said.
The officer saw something moving, got out of his car, saw Clemmons and ordered him to show his hands and stop.
"He wouldn't stop," Pugel said. "The officer fired several rounds."
Clemmons also had sustained a serious gunshot wound from one of the four officers killed in the coffee-shop shooting.
Police planned to arrest more people who helped Clemmons.
"We expect to have maybe six or seven people in custody by the day's end," said Ed Troyer, a spokesman for the Pierce County sheriff. "Some are friends, some are acquaintances, some are partners in crime, some are relatives. Now they're all partners in crime."
Three people were booked into the Pierce County Jail on Monday and early Tuesday for investigation of rendering criminal assistance on four counts of first-degree murder. They are Ricky Hinton, Eddie Lee Davis and Douglas Edward Davis. Troyer said a getaway driver also was arrested. That person's identity wasn't immediately known.
On Monday, officers detained a sister of Clemmons who they think treated the suspect's gunshot wound.
"We believe she drove him up to Seattle and bandaged him up," Troyer said.
Authorities say Clemmons, 37, singled out the Lakewood officers and spared employees and other customers at the coffee shop Sunday morning in Parkland, a Tacoma suburb about 35 miles south of Seattle. He then fled, but not before one of the dying officers apparently shot him in the torso.
"I'm surprised that he managed to get away," Troyer said. "The officer did a good job in Lakewood."
Killed were Sgt. Mark Renninger, 39, and Officers Ronald Owens, 37, Tina Griswold, 40, and Greg Richards, 42.
A couple dozen police officers milled around at the scene where Clemmons apparently was shot, shaking hands and patting each other on the back later Tuesday morning. The officer who shot Clemmons was not injured, Pugel said.
Police said they aren't sure what prompted Clemmons to shoot the four officers, who were in uniform and working on paperwork at the coffee shop just two blocks outside their jurisdiction.
"The only motive that we have is he decided he was going to go kill police officers," Troyer said. He said Clemmons talked the night before the shooting about killing a group of cops and watching the news.
Police believe Clemmons chose the coffee shop because it was frequented by police officers from various agencies.
"We do not believe that the Lakewood officers were actually targeted other than that they were police officers in that location at the time where he knew he could find police officers."
Clemmons was described as increasingly erratic in the past few months and had been arrested earlier this year on charges that he punched a sheriff's deputy in the face.
Police surrounded a house in a Seattle neighborhood late Sunday following a tip Clemmons had been dropped off there. After an all-night siege, a SWAT team entered the home and found it empty. But police said Clemmons had been there.
Police frantically chased leads on Monday, searching multiple spots in the Seattle and Tacoma area and at one point cordoning off a park where people thought they saw Clemmons.
Authorities found a handgun carried by the killer, along with a pickup truck belonging to the suspect with blood stains inside. They posted a $125,000 reward for information leading to Clemmons' arrest and alerted hospitals to be on the lookout for a man seeking treatment for gunshot wounds.
Authorities in two states were criticized amid revelations that Clemmons was allowed to walk the streets despite a teenage crime spree in Arkansas that landed him an 108-year prison sentence. He was released early after then-Gov. Mike Huckabee commuted his sentence.
Huckabee cited Clemmons' youth in granting the request. But Clemmons quickly reverted to his criminal past, violated his parole and was returned to prison. He was released again in 2004.
"This guy should have never been on the street," said Brian D. Wurts, president of the police union in Lakewood. "Our elected officials need to find out why these people are out."
Huckabee said on Fox News Channel's "The O'Reilly Factor" Monday night that Clemmons was allowed back on the street because prosecutors failed to file paperwork in time.
Pulaski County Prosecutor Larry Jegley, whose office opposed Clemmons' parole in 2000 and 2004, said Huckabee's comments were "red herrings."
"My word to Mr. Huckabee is man up and own what you did," Jegley said.
Clemmons was charged in Washington state earlier this year with assaulting a police officer and raping a child, and investigators in the sex case said he was motivated by visions that he was Jesus Christ and that the world was on the verge of the apocalypse.
But he was released from jail after posting bail with the assistance of Jail Sucks Bail Bonds.
Documents related to those charges indicate a volatile personality. In one instance, he is accused of gathering his wife and young relatives and forcing them to undress.
"The whole time Clemmons kept saying things like trust him, the world is going to end soon, and that he was Jesus," a Pierce County sheriff's report said.