It’s still going to take a battle for the U.S. women’s national soccer players to be paid on some sort of an equal basis as the men, but progress is being made, one prominent member of the team says.

Christen Press said the support the team received not only for its march to the World Cup championship in France but for its push for equal pay was “jaw-dropping.”

“The lawsuit (against the U.S. Soccer Federation) and the whole process has been a fight,” the Stanford alum said in a phone interview Tuesday. “We’ve learned to fight for our worth and to accept nothing less. The World Cup has been a celebration of all the progress we’ve made and of how far the game has come.”

She was speaking from Salt Lake City, where she and USA teammates Kelley O’Hara (another ex-Cardinal player) and Becky Sauerbrunn rejoined the Utah Royals of the National Women’s Soccer League this week.

They received the keys to the city, which they can place on their trophy shelf next to the keys to the city of New York, where they were honored in a ticker-tape parade on Broadway’s “Canyon of Heroes” on July 10.

Since beating the Netherlands 2-0 to win the team’s fourth World Cup title July 7, the women have had whirlwind trips around the country.

“It’s just been crazy,” Press said. “We were living out of a suitcase for two months.”

After New York, she and the team flew to Los Angeles, where they were honored as the best team — in any sport — at the ESPYs.

Press also presented the Stanford women with the Capital One Cup (given to the nation’s top men’s and women’s college athletic programs). The 2010 Hermann Trophy winner as the best women’s player in the nation, Press holds the Cardinal’s career records of 71 goals and 183 points.

She also spent a couple of days kicking back in Mexico before flying home to Portland.

Press, 30, played a big role in putting the U.S. in the title game. Starting for the injured Megan Rapinoe, she scored the first goal in a 2-1 semifinal win over England.

“I felt very prepared and very confident and very supported by my teammates, especially Megan,” she said. “I think that (support) was what contributed to our overall success.”

It was her second straight World Cup championship. But after the 2015 team won the Cup in Canada, it lost in the Olympic quarterfinals to Sweden the next year. Press missed the team’s last kick in a 4-3 penalty shootout.

“I don’t use words like ‘get over’ because life is so fluid,” she said about getting past the misfire. “The time after that Olympics was probably the time I’m most proud of in my life. ...

“So many people have overcome huge losses. I’m proud of that time. It’s not easy to keep going and to continue to be creative (on the field) and not back down, to say, ‘I’ll take another penalty’ and not let that moment define you. But I have control over my own narrative, and I wasn’t going to let that moment define my narrative. I think I did that. It will always be a part of me, but actually a part that I’m quite proud of.”

The Olympics disappointment helped get the U.S. the World Cup trophy three years later, she said.

“We learned our lessons in 2016, and that allowed us to prepare in a better way for the 2019 World Cup,” she said. “We’ll use more of these lessons going into the upcoming Olympics.”

Tom FitzGerald is a San Francisco Chronicle staff writer. Email: tfitzgerald@sfchronicle.com Twitter: @tomgfitzgerald