By Jim Forsyth

SAN ANTONIO (Reuters) - An Army psychiatrist accused of killing 13 people in a shooting rampage at Fort Hood, Texas, in 2009 could learn on Tuesday whether a judge will grant him more time to prepare as he represents himself in the military trial.

Major Nidal Hasan, 42, could face the death penalty in the trial, now scheduled to begin with opening statements on July 1 after several delays, most recently because he has been allowed to represent himself and relegate his military lawyers to an advisory role.

The judge, Colonel Tara Osborn, has been pressing to get the court martial back on track nearly four years after the November 2009 attack on a facility where soldiers were preparing to deploy to Iraq and Afghanistan. Thirteen people died and 32 were wounded.

Fort Hood was a major deployment point for the wars in Iraq and Afghanistan, and Hasan himself had been preparing to leave for Afghanistan with a unit assigned to help soldiers deal with mental issues.

Among the unresolved issues: How will potential jurors be questioned, what new role should the defense attorneys play now that they are not leading his defense, and how to talk to a jury of military officers about Hasan's beard.

A U.S.-born Muslim, Hasan says he wears a beard for religious reasons, although that violates military dress code.

Military officials at the Texas Army post were not specific about which legal questions Osborn would answer on Tuesday.

Hasan, who was shot by civilian base police during the attack, unsuccessfully sought to argue at trial that he was protecting the Taliban from American aggression. Last week, Osborn denied a request to argue that line of defense.

Hasan, now paralyzed from the chest down, has asked for three more months to develop a new defense strategy and witnesses. Experts said Osborn could give Hasan additional time to protect the integrity of a future verdict.

"If I were a betting man, I would say maybe a delay of a week," said former Army prosecutor Geoffrey Corn. "This will eliminate the risk that, on appeal, some appellate court will say you should have granted a delay."

Osborn decided to ignore Hasan's beard - an issue that delayed proceedings for months and caused another judge to be removed from the case - but may need to address jurors about it.

(Editing by Karen Brooks)