In the video, Stephens claimed to have killed more than a dozen people, police said, though they have not confirmed any other victims.
While authorities said Stephens might have left OH, the city's police chief added that his last known location was at the site of the shooting Sunday afternoon and that detectives spoke with the suspect by cellphone.
Cleveland police chief Calvin Williams says authorities have already searched every location Stephens has been associated with in the Cleveland area.
In another video posted to Facebook, Stephens said that he gambled away everything and that he and his girlfriend had planned to marry but didn't, without saying why.
The victim was 74-year-old Robert Godwin, a father-of-nine, who was walking along a road in Ohio, USA. He had been collecting cans on his walk, and police believe he was randomly targeted.
As of Monday at 1 p.m., the campaign had raised $24,000.
A former girlfriend of the suspect in the slaying of an elderly Cleveland man has told CBS News in a text that 37-year-old Steve Stephens was "a nice guy" who was generous to everyone and was "kind and loving" to her and her children. I am sorry that all of this has happened.
Police have offered a $50,000 reward for information leading to Stephens' capture.
"We are shocked and horrified like everyone else", said Nancy Kortemeyer, a spokeswoman for Beech Brook.
Police Chief Calvin Williams warned residents to be careful as the go about their day. He was last known to be driving a White Ford Fusion with the temporary OH license number E363630. He was last seen with a full beard and wearing a dark blue and gray striped shirt, police said.
Hillary Clinton apologized to Obama on election night
She said that it isn't helpful for Democrats to look back at the election, when they should be looking forward. The call worked and Mrs Clinton's close aide Huma Abedin called Mr Trump's campaign manager Kellyanne Conway.
"She's the reason why this is about to happen to you", the gunman said.
"As a result of this awful series of events, we are reviewing our reporting flows to be sure people can report videos and other material that violates our standards as easily and quickly as possible", Justin Osofsky, a Facebook vice president, said in a blog post.
Williams said Stephens may be driving a newer model white Ford Fusion.
Air force veteran Ms Lane said: "We had been in a relationship for several years". "We work hard to keep a safe environment on Facebook, and are in touch with law enforcement in emergencies when there are direct threats to physical safety".
The company said it disabled Stephen's account within 23 minutes of receiving the first report about the video of the fatal shooting and two hours after receiving any report. It is believed the video was on Facebook for three hours before it was deleted.
One rumor on our Facebook page is that his cell phone pinged in Erie, Pa. Police in the city of Erie - and the state police - tell us these reports are also not true.
Williams said his department and federal agents have searched dozens of locations around Cleveland and have yet to locate Stephens. "To think that one of our employees could do this is bad".
Stephens worked at Beech Brook, a social services agency in suburban Cleveland that deals with vulnerable young people.
Alexis Lee, who saw Stephens last week, said his childhood friend always seemed respectful and got along with everybody.
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