In addition to the Air Force investigation, the Pentagon’s inspector general is looking into the handling of Mr. Kelley’s court-martial conviction records and whether procedures are in place to ensure cases from other service branches are also sent to the database as required. The Army chief of staff, Gen. Mark Milley, said that he believed there were also a significant number of omissions of soldiers’ records that should have been sent to the federal database.

And the number of cases that were not properly reported by the Air Force could grow: There have been about 60,000 incidents in the Air Force since 2002 involving service members that potentially should have been reported to the federal background-check database. All of those incidents are now being reviewed by Air Force officials to see which ones were required to be reported, and how many of those actually were. Air Force officials were unable to say on Tuesday how many of those 60,000 cases have gone through the review process so far.

While that review is expected to continue for several months, Air Force officials say they are moving to fix problems that prevented Mr. Kelley’s conviction from being reported. And officials emphasized that they would continue to send previously unreported cases to the federal database as soon as they discover them.

Among other things, the new directive includes a requirement that personnel at the Air Force Office of Special Investigations must confirm that reportable cases have been entered into the federal database by seeing either a printout or a screenshot from the database.

Don Christensen, who was the chief prosecutor in the Air Force and is now president of Protect Our Defenders, a nonprofit group that supports greater protections for victims of sexual assault and domestic violence in the military, said the measures were long overdue.

“I’m not surprised that they are finding these lapses, because this was clearly never a priority in the past,” Mr. Christensen said. “Earlier inspector general investigations found that they were not doing this properly, and the leadership never made it a priority to correct it.”

Even if Mr. Kelley’s conviction had been in the federal database, he could still have purchased a gun online or in person from private sellers not required to run a background check — an exemption known as the “gun show loophole.”