BATON ROUGE, La. — The Pete Maravich Assembly Center has spent 50 years seeing commencements and concerts, presidential power and a governor’s prayer rally.

It was not until this past weekend, though, that it came within about five minutes of hosting the second-biggest upset in the history of the N.C.A.A. women’s basketball tournament.

Then the No. 3 seed Louisiana State surged, 14th-seeded Jackson State sputtered and the tournament, long derided as something of a predictable spectacle, went on apace. But the Southern drama — and the theatrics that emerged elsewhere, in places like Iowa City and College Park, Md. — showed how the competition can yield stunners, near misses and, as expected, mostly unchecked and unchallenged marches by the sport’s elite toward next month’s Final Four.

And escaping one game is, as ever, no guarantee of surviving the next: L.S.U. lost to Ohio State, seeded sixth, on Monday night. The reigning national runner-up went down, too, and Connecticut, which has won 11 championships, nearly did.

“There are enough good players out there that everybody can be successful,” Dawn Staley, South Carolina’s coach, said before the field shrank to 16 teams on Monday night. “I’m not really surprised by some of the mid-majors being able to upset some higher seeds.”

She added: “All of our game is pretty special, and now we’re just really starting to see the other side of it. I think sometimes we just concentrate on the hot teams and coaches in the Power 5. Now the mid-majors are, you know, they’re rising. I don’t think they’re catching anybody off guard.”

Starting on Friday, the tournament, which began with 68 teams, its largest field ever, will move from the home courts of the top 16 seeds to regional sites in Bridgeport, Conn., Greensboro, N.C., Spokane, Wash., and Wichita, Kan. The Final Four will be played in Minneapolis on April 1, with the championship game tipping off there two days later.

But after four play-in games and two full rounds, no one can yet feel fully safe. In fact, for some, the perils are only beginning.

South Carolina, the tournament’s top overall seed, will next face North Carolina, which on Monday pummeled last season’s runner-up, Arizona. North Carolina State, another No. 1 seed, has a path to Minneapolis that may involve having to beat the University of Connecticut, which is seeking its 14th consecutive Final Four berth but looked shaky on Monday, in Connecticut. Louisville could wind up facing South Dakota, which has already ousted second-seeded Baylor and Mississippi, a No. 7 seed, from the tournament. If Stanford, the top seed in the Spokane region, and Ohio State meet, one bench would including the 2021 champion and the other a program that recently knocked off a title-winning coach’s latest team.

This year’s men’s tournament has the feel of a wide-open frenzy. Four of the remaining teams are double-digit seeds. Some of the sport’s historically great powers — including Duke, Kansas, North Carolina and U.C.L.A. — are still in contention. Gonzaga, the tournament’s top overall seed and the kind of program that seems like it to have won a title sometime, is pursuing its first one.

The women’s tournament is still chiefly the domain of a narrow slice of Division I basketball. It is not, however, as monolithic as it once was. The last five tournaments have yielded five champions, none of whom shared a conference, moving the sport away from UConn’s dominant era.

That is helping to fuel greater interest in the tournament, which is enjoying the N.C.A.A.’s “March Madness” branding for the first time after the association’s self-inflicted firestorm last year about gender equity in college basketball.

ESPN said the television broadcast that debuted this season’s women’s bracket drew more than 1.1 million viewers, the program’s best showing since 2005. ESPN also said that submissions for its women’s bracket contest had increased 67 percent from last year. Face-value tickets for the title game appear to be selling well. (The secondary market, perhaps enlivened by the departures of Baylor and Iowa, has offerings, too.)

The games this week and early next, though, are likely to be the exit ramps for the merely lucky.

“It’s settling in that there is only 16 teams left playing in the tournament,” said Liv Korngable, a redshirt senior guard for South Dakota, “so it’s very cool.”

Her team, a No. 10 seed with a habit of defanging ordinarily potent offenses, will play Michigan, which earned a No. 3 seed, on Saturday.

But the route to the Final Four for either could very well go through Louisville, the Wichita region’s top seed. The Cardinals, who will be making their 11th appearance in the round of 16, will square off on Saturday against Tennessee, which held off Belmont on Monday.

“I see us going to a championship, but at the same time you’ve got to make sure you don’t make, two, three, four, five mistakes,” said Emily Engstler, a Louisville senior. “That’s what we’re trying to do: We’re trying to limit our turnovers, keep playing really high intensity defense, staying aggressive and composed.”

South Carolina will play close to home on Friday, when it will anchor the Greensboro region. Even the Gamecocks, who had a halftime lead of 40 points in their first tournament game, have shown vulnerability: They initially struggled on Sunday to overcome a swarming Miami defense that rattled Aliyah Boston, a sensational forward, and held her to 10 points over 35 minutes. She had scored that many in 18 against Howard.

And the Tar Heels will be meeting the Gamecocks on a court even closer to their campus.

Also in the Greensboro region, Creighton could become the de facto, invading champion of the Hawkeye State if, after it edged second-seeded Iowa on Sunday, it can similarly thwart Iowa State, a No. 3 seed. Like South Dakota, Creighton thrived over the weekend with resounding defense, holding Caitlin Clark, the Iowa sophomore who led Division I in scoring this season, to a season-low 15 points.

Stanford hardly seemed to sweat in its first two games, which it won by an average of about 34 points. Then again, Maryland, its opponent this Friday, barely seemed to break a sweat itself. (It prevailed by merely 27.5 points per game.)

Ohio State, among the country’s best programs behind the arc, will meet a feisty Texas team — “Punch first, that’s always our motto,” Aliyah Matharu, a junior guard, said — that did not allow more than 56 points in either of its first two tournament games but is still carping over its communication on defense.

“One game at a time — that’s what we put in our team group chat,” said Matharu, whose team is a No. 2 seed. “We want to do one game at a time. We’re not going to think too far ahead even though you see a lot of upsets. It’s important to stay within us. It’s about our team, our teammates, each other, and I think if we do that, then we’ll be fine.”

The most seismic game of the next week may come on Monday, when N.C. State and UConn could meet in a regional final in Bridgeport. It has been a trying season for UConn, which has struggled with injuries and beat Central Florida by just 5 points on Monday, and N.C. State, which has not advanced beyond the round of 16 since 1998, is looking to prove that it can be more than a regular-season titan.

N.C. State will first have to maneuver around No. 5 Notre Dame, which on Monday demolished No. 4 Oklahoma, 108-64. And UConn will have to get past No. 3 Indiana, which defeated Princeton by only a point on Monday.

Teams from across the country, meanwhile, will keep peering up and down the brackets, looking to see which opponents might emerge from the chaos.

“We never look ahead,” Tara VanDerveer, Stanford’s coach, preaches to her team, “but we have to plan ahead.”