The sheriff said deputies chased the man to a nearby business where they fired a stun gun at the escapee and took him into custody. He was returned to the jail.
Jondreau suffered a broken heel bone and had multiple cuts from razor wire, the sheriff said. His cuts required about 80 sutures.
Jondreau is charged in Williams County, Ohio, with killing his 9-week-old son, the sheriff said. The Polk County Sheriff's Office arrested him on an out-of-state warrant on May 11, and he was transferred to Pinellas County on May 28 to await extradition to Ohio.
Gualtieri said Jondreau will be transferred to Ohio once he's medically cleared. He said authorities will ask the state attorney's office to charge the man with with escape once he leaves the Pinellas County Jail.
The sheriff said jail officials will review safety protocols.