6:58 AM PT -- TMZ Sports has obtained the 911 call placed by Smith's wife from the school ... and she says Zach was irate -- "yelling" at the principal until cops arrived.

More trouble for former Ohio State coach Zach Smith ... cops say they arrested Urban Meyer's ex-assistant after he got into an altercation with officials at his children's elementary school Thursday.

The Delaware County Sheriff's Office confirms to TMZ Sports that Smith showed up to Scioto Ridge Elementary School in Ohio to pick up his kids at around 3 PM.

The problem? They say because of a civil protection order with his wife -- who was already at the school -- Smith was NOT allowed to be there until 6 PM.

When school officials confronted Zach ... cops say an altercation took place -- and Zach was eventually arrested for violating the protection order.

Per jail records, Smith was booked at 6:24 PM and has not been released yet.

As we previously reported ... it's been a nightmare past year for Smith -- who was fired from his role as Buckeyes' receivers coach last summer when he was the centerpiece of a domestic violence scandal involving his wife, Courtney.

She accused him of abusing her on several occasions ... and while Zach denied ever getting violent, he admitted things had gotten physical during heated disputes in the past.

The 35-year-old has been out of coaching since his Ohio State departure in July.