Eddie Ray Routh, 25, is accused of the shootings at a gun range on Saturday.
Sheriff Bryant said Mr Routh then travelled to his sister's home, telling her what had happened before leaving.
BBC: Chris Kyle murder: Suspect Eddie Ray Routh subdued in jail
Singer's new film stars Brandon Routh, who takes over the role following the death of Christopher Reeve in 2004.
BBC: Superman actor Routh was chosen from hundreds of hopefuls
Afterward, Littlefield's relatives said the outing with Routh was intended to be therapeutic.
Kyle and LIttlefield, friends and associates, arrived with Routh at the sprawling range southwest of Fort Worth together, police said.
Police finally caught up with Routh near his home in Lancaster, about 15 miles south of Dallas, around 8 p.m.
Erath County Sheriff Tommy Bryant said Mr Routh then travelled to his sister's home, telling her what had happened before leaving.
An Iraq War veteran, Eddie Ray Routh, 25, has been charged in the killings of Kyle and his friend Chad Littlefield.
When odd or embarrassing things happened to people after 1960, they would half-expect Mr Routh and his camera to be watching.
Routh's brother-in-law told authorities that Routh had recently been diagnosed with PTSD.
By creating a digital version of Routh, animators were able to make him fly and do all kinds of things humans can't do.
Mr Kyle and his neighbour Chad Littlefield were reported to have been trying to help Mr Routh deal with post-traumatic stress disorder (PTSD).
By then, Routh had taken off in Kyle's black Ford pickup, stopping first at his sister's house about 70 miles away in Midlothian.
It also boasts a strong cast, with Routh and Bosworth's relative inexperience balanced by the seasoned likes of Frank Langella and Eva Marie Saint.
After the shooting at Rough Creek Lodge range on Saturday afternoon, Mr Routh allegedly fled in a pick-up truck belonging to one of the victims.
Police arrested suspect Eddie Ray Routh, 25, after a brief pursuit and arraigned him on two counts of capital murder, according to the safety agency.
Mr Routh told police he suffered from post-traumatic stress disorder (PTSD).
Records show Mr Routh was re-admitted to the same hospital in January after a woman told the police she was afraid the young man might harm himself.
She called the police and Mr Routh was arrested several hours later near his home in Lancaster, Texas, more than 70 miles (110 km) from the crime scene.
BBC: Chris Kyle murder: Suspect Eddie Ray Routh subdued in jail
Authorities say Kyle, Littlefield and Routh arrived together at the Rough Creek Lodge shooting range, about 50 miles southwest of Fort Worth, on the afternoon of Feb. 2.
But what Jonathan Routh preferred to do was to dress up as a tree, wait at a bus stop and enquire which bus would take him to Sherwood Forest.
She called the police and Mr Routh was arrested near his home in Lancaster, Texas, more than 70 miles (110 km) from the scene of the shooting, several hours later.
On Sunday night, authorities said they had to use a stun gun to subdue Mr Routh in custody after he became belligerent and refused to hand back his food tray.
Erath County Sheriff Tommy Bryant said Routh served in the Marines for four years, but it was unclear how much of that time, if any, was spent in combat zones.
According to police reports in Lancaster, Texas, where Mr Routh lives, the veteran was admitted to hospital for psychiatric treatment on 2 September after threatening to kill his family and himself.
The club is a creation of Broadway producers Tom Viertel, Marc Routh, Richard Frankel and Steven Baruch, who were looking to create a place for smaller-scale performances rather than full-blast musical theater.
"Man of Steel, " which replaces Brandon Routh with Henry Cavill, is the origin story of Superman, starting with a young Clark Kent discovering that he has extraordinary powers and is not from planet Earth.
Travis Cox, director of a non-profit organisation Kyle had helped found, told the Associated Press news agency that Mr Routh was suffering from PTSD and that Kyle and Littlefield were trying to help him.
BBC: Chris Kyle murder: Suspect Eddie Ray Routh subdued in jail
Travis Cox, director of a non-profit organisation Mr Kyle had helped found, told the Associated Press news agency that Mr Routh was suffering from PTSD and that Mr Kyle and Mr Littlefield were trying to help him.
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