Three people were killed and 16 others wounded in the shooting spree before the suspect, Spc. Ivan Lopez, killed himself, authorities said.
During a news conference Monday, Army spokesman Chris Grey said the shootings at the Texas post followed an argument related to Lopez's request for taking leave, but he didn't indicate whether it was granted or describe circumstances behind the request.
A spokesman for Lopez's family said last week that Lopez was upset he was granted only a 24-hour leave to attend his mother's funeral in November. That leave was then extended to two days.
The shooting spree Wednesday ended when Lopez killed himself with his .45-caliber pistol after confronting a female military police officer, who Grey said fired once at Lopez but didn't strike him.
Providing the most detail yet about the second mass shooting at Fort Hood in five years, Grey mapped out how Lopez opened fire in the building where the argument began before leaving and driving away, shooting at times from his car. The three who died were gunned down in separate locations.
Grey said Lopez first killed one soldier and wounded 10 others in the first building - the dead being one of the men Lopez had argued with moments earlier. Lopez then drove to a motor pool area where the Army truck driver was assigned and worked, killing another, Grey said.
The last place Lopez entered was a block away at a medical building, Grey said, walking inside and killing a soldier behind the desk.
"At this point we do not know why he entered that building, and we may never know why," Grey said.
In all, investigators say Lopez fired more than 35 shots.
Authorities said 11 of 16 injured have returned to duty. Three soldiers who remain at nearby Scott & White Hospital, where the most critically injured were taken, were listed in fair condition Monday.
President Barack Obama is scheduled to attend a memorial service Wednesday at Fort Hood. Authorities said transportation arrangements for the three dead are being finalized for their funerals.
In another attack at the base in 2009, 13 people were killed by Army psychiatrist Nidal Hasan, who had said he was angry about being deployed to Afghanistan and wanted to protect Islamic and Taliban leaders from U.S. troops.
Lopez did a short stint in Iraq in 2011 and told medical personnel he had suffered a traumatic brain injury. The 34-year-old was undergoing treatment for depression and anxiety while being evaluated for post-traumatic stress disorder, base officials said. Fort Hood officials on Friday, however, said his mental condition was not a "direct participating factor" in the shooting.
Officials said Lopez did not see any combat in Iraq and had not previously demonstrated a risk of violence. He seemed to have a clean record and Grey said again Monday that Lopez showed no ties to potential terrorists.