The chief said detectives are looking to talk to Chimene Onyeri to see whether he has any connection with Judge Julie Kocurek's assault.
"Every possible avenue is being explored in an attempt to include or exclude the person that we have as a person of interest in an attempt to make sure we bring to justice everybody that was involved," he said.
Acevedo was asked by CNN whether investigators could place Onyeri in the Austin area Friday night. The chief didn't answer the question, saying he did not want to share details of the investigation.
Kocurek, the presiding felony judge for Travis County, was attacked in her driveway as she was driving home from a football game.
Court records show that Kocurek issued a warrant for Onyeri's arrest on August 28 in a gift card fraud case. He was due to appear before her in December and was facing possible prison time after violating his three-year probation.
Acevedo said Austin detectives are in Houston to question Onyeri, 28, about the judge's shooting, but it was unclear whether that had begun.
Acevedo wouldn't comment on the judge's condition, but said, "We're very hopeful that she will recover."
When asked whether there were other persons of interest in Kocurek's shooting, Acevedo said the investigation isn't close to being wrapped up.
Onyeri father, Innocent Onyeri, told CNN affiliates KPRC
in Houston that his son was at his home all weekend.
Earlier Tuesday, Houston police said Onyeri is accused of killing Jacobi Demond Alexander, 31, on May 18. Onyeri, one of two suspects, according to police, was arrested Monday in Houston.
He has a preliminary hearing scheduled for Thursday morning, and online court records didn't indicate that he had an attorney. CNN's attempts to find an attorney for Onyeri were unsuccessful.
Attacked in her driveway
The Austin American-Statesman
, citing sources, said someone put a garbage bag or can in the middle of the judge's driveway Friday night, forcing Kocurek's car to stop.
Someone came out of the darkness and fired.
Kocurek's injuries were from shrapnel and glass, the newspaper said.
Colleagues wondered whether it was a targeted attack.
"The idea that you would be ambushed or attacked in your home for the work that you do was just mind-boggling to me," Travis County Administrative Judge Lora Livingston told CNN.
'She would have had to make some enemies'
Investigators are poring over the judge's extensive cases. Kocurek has spent about 25 years in Austin working as a prosecutor and then a judge.
Defense lawyer David Minton said there's no question in his mind the judge was targeted. He has known Kocurek for years and says she's long been considered a fair and thoughtful judge.
"But clearly in her job and the number of years she's been on the bench, she would have had to make some enemies for doing her job," he told CNN. "It's just something that comes with the territory. But you never think it would come to this."
Kocurek has been the presiding judge of the 390th District Court since January 1999, according to her official webpage
. She's had her hand in many cases in that time, including high-profile political cases involving former Texas Gov. Rick Perry
and former U.S. House Majority Leader Tom DeLay
She has also been active in the local community, including serving as a local co-chair in the National Campaign to Stop Violence
, a past board member of the Mothers' Milk Bank at Austin
and a supporter of the West Austin Youth Association
In recent years, several judges and prosecutors have been targeted in the United States.
U.S. District Judge Richard Daronco was shot and killed in 1988 at his Pelham, New York, home by a man upset about Daronco's dismissal of the man's daughter's sexual discrimination lawsuit.
The next year, federal Appeals Court Judge Robert Vance died after opening a mail bomb.
And in 2005, the husband and mother of U.S. District Court Judge Joan Humphrey Lefkow were shot dead inside Lefkow's Chicago home
. A plaintiff whose case Lefkow had dismissed claimed responsibility for the slayings in a suicide note.
More recently, and closer to Kocurek's Austin home, prosecutor Mark Hasse was fatally shot
in 2013 outside the Kaufman County, Texas, Courthouse.
Kaufman County District Attorney Mike McLelland responded to that shooting by vowing to put away the "scum" who'd killed one of his top deputies. Two months later, McLelland and his wife, Cynthia, were found shot dead
inside their home east of Dallas.
Eric Williams, a former Kaufman County justice of the peace, was later convicted of capital murder
in Cynthia McLelland's death. Williams wasn't put on trial for the death of Michael McLelland or Hasse, though authorities did accuse him in both killings.