President Trump is going old-school with the death penalty ... approving execution by firing squad for the man convicted in the 2009 Fort Hood massacre.

Trump signed off on the execution of former Army Maj. Nidal Malik Hasan, who was sentenced to death for killing 13 people and wounding more than 30 others.

The move comes just days after Tennessee's attempt to execute Christa Pike by lethal injection went horribly wrong ... with Pike surviving two doses of pentobarbital before being rushed to the hospital.

As TMZ previously reported, Tennessee Gov. Bill Lee halted the state's remaining executions for the year after the botched attempt and ordered an outside review.

Hasan's execution would make history ... the U.S. military hasn't put anyone to death since 1961, when Army Pvt. John Bennett was hanged at Fort Leavenworth ... and the last military firing-squad execution was even longer ago ... Pvt. Eddie Slovik was shot to death in France in 1945 for desertion during World War II.

Hasan admitted at trial he was the gunman, though he said he carried out the attack to stop U.S. soldiers from harming Muslims overseas.

Now, while Tennessee is scrambling to figure out what went wrong with lethal injection ... Trump is backing a much different method for one of the country's most notorious military killers.