LINDEN, Texas—Attorneys representing the man accused of killing a woman and her two young daughters in their Atlanta, Texas, mobile home in 1992 asked the judge to approve the appointment of another lawyer to help with the defense.

Michael Kevin Hailey, 40, is facing the death penalty if found guilty of committing the 1992 murders of Jerri Faye Butts, 29, and her daughters, Jessica Butts, 11, and 3-year-old Mackenzie Sullivan Butts.

Friday morning, Hailey’s lawyers, Mitch Collins and Brian Simmons of the Collins Law Firm, asked Fifth District Judge Ralph Burgess to appo...