South Carolina police think a shootout occurred before constable Robert Lee Bailey may have been taken somewhere against his will earlier this week, police told FOX News Thursday.

Children unwrapping an abandoned wad of burned rags on Wednesday discovered the missing constable's badge, handcuff case and other items, authorities said. There was still no sign of Bailey on Thursday.

Authorities have arrested three people they hope will provide information on the officer's disappearance; Bailey, 67, went missing after a routine traffic stop while patrolling the small community of Lincolnville Monday night.

On Wednesday, authorities arrested Walter Fayall III on federal weapons charges and called him a "person of interest" in Bailey's disappearance, but no charges have been filed in the officer's case.

Charleston County Sheriff J. Al Cannon told FOX News on Thursday that Fayall does have a criminal record that includes car theft, but nothing like murder or kidnapping.

Authorities are investigating whether Fayall, 24, lives in a house in Lincolnville near where gunshots were heard about the time Bailey disappeared.

One of the other men, Jerome Washington, 26, of Pineville, was charged Wednesday with arson and trespassing in an unrelated incident in Williamsburg County, said Michael Johnson, operations director at the county jail. Johnson said U.S. marshals picked up Washington on Thursday and took him back to Charleston.

Also, Brian Smalls was being held in a Berkeley County jail on a probation violation.

No charges related to Bailey's disappearance had been filed by Thursday afternoon.

"I feel good about where we are in terms of the investigation," the sheriff said.

Police are "looking closely at" these individuals for possible connections to the Bailey case, Cannon told FOX News. Police have already discounted other potential suspects after their alibis checked out.

"We feel pretty confident about what we've developed to this point," Cannon told FOX News. "We're real confident we're on to the right people."

About 75 officers were searching the rural Harleyville area about 45 miles northwest of Charleston for any sign of Bailey. The retired Dorchester County sheriff's deputy was last seen at a traffic stop late Monday. His hat, gun, blood and several bullet casings were found in a yard, and his empty cruiser was found torched behind a church off Interstate 26, police said. Shell casings appear to have come from two guns, one of them Bailey's, Cannon told local news outlets.

Earlier in the evening, Bailey spent about 10 minutes conducting a routine traffic stop, which he called in to police headquarters. He had just left that stop when he came upon whatever caused him to disappear, Cannon said.

"I don't think it was a typical traffic stop," Cannon told FOX News.

"Within just a couple of minutes, he ran up on something that happened so quickly, he didn't have time to radio it in or whatever. We're at a loss at what exactly happened ... but something happened very quickly. There was an exchange of gunfire," and authorities believe Bailey was then taken somewhere, along with his police cruiser.

The Charleston Post and Courier reported that it was about 10:30 p.m. Monday when Sgt. Richie Hill, the only paid police officer in Lincolnville, heard the shots from his office. He knew Bailey, a volunteer police officer, had been on a traffic stop just down the road, so he radioed the dispatcher to check on him, Capt. Mike Benton of the Charleston County Sheriff’s Office told the newspaper. But the constable didn’t answer the radio.

Click here to read The Charleston Post and Courier story

Shortly after the routine traffic stop, 911 calls rang in to the sheriff’s office from residents concerned about gunfire, the newspaper reported.

"As time goes on, it's certainly more bleak," Cannon told FOX News. "Originally we found a great deal of blood at the scene ... we think some shootout occurred."

Tests have not yet concluded whose blood it was.

The Associated Press contributed to this report.