She is the unlikeliest player in a most improbable last four but Sloane Stephens’ run to the business end of this year’s US Open feels in many ways right on schedule.

The 24-year-old is one of four American women through to Thursday night’s semi-finals after the top eight women’s seeds were eliminated by the quarters or earlier for the first time at a grand slam in the Open era. None of them, least of all the unseeded and unfancied Stephens, were favored to make it this far.

Now Stephens, the world No83, is one victory away from her maiden grand slam final. With a win over Venus Williams at Arthur Ashe Stadium, she would become only the fourth unseeded player to reach the US Open final since the majors allowed professionals to compete with amateurs in 1968.

Few could have imagined it at the start of August, when the Florida native was down to 957th in the world entering a tournament in Washington DC where she lost in the opening round. Stephens was still finding her form after a season-ending stress fracture last year that required surgery and an 11-month layoff, including nearly four months on a self-described “peg leg” when she was unable to put weight on her foot.

But Stephens has since won 13 of 15 matches, all against opponents ranked in the top 50, making the semi-finals at high-profile events in Toronto and Cincinnati and, now, the US Open. She will be ranked no lower than 35th when Monday’s rankings are released after reaching her second career major semi-final – a jump of more than 900 spots in less than two months.

“I said eventually: ‘I’m going to beat somebody and them I’m going to beat two people in a row,’” Stephens said after Tuesday’s quarter-final upset of the No16 seed, Anastasija Sevastova. “And then did I think it was going to be the next week? No. I just stayed positive. And then, yeah, look, there, semi-final, semi-final, semi-final. I couldn’t really ask for a better way to come back.”

Positivity was the hallmark of the first coming of Stephens, who first cracked the top 100 with a precocious third-round run at the 2011 US Open and was still a teenager when she first made the second week of a grand slam one year later at Flushing Meadows, featuring heavily in the advance promotion of the season-ending grand slam.

When Stephens stunned Serena Williams in the 2013 Australian Open quarter-finals, it was the first time Williams had ever lost to an American younger than herself. And when Stephens backed it up with runs to the French Open fourth round and Wimbledon quarter-finals in the same year to soar to No11 in the rankings, she was broadly cast as the next big thing in American tennis.

All of it was standard achievement in a family of gifted athletes: her mother, Sybil Smith, was an all-American swimmer at Boston University in the late 1980s, while her father, John Stephens, was a Pro Bowl running back for the New England Patriots who died in a car accident in 2009. But Sloane, whip-smart and engaging with a megawatt smile made for Madison Avenue, threatened to outstrip both their achievements.

However the role of successor to the Williams sisters brought with it a whole new array of pressures and expectations, inviting a dip in form and dulling her appreciation for the sport over time. The foot injury, a result of rushing back to the tour after jetting home to spend time with her ailing grandmother, was at the time yet another unfortunate break. But Stephens credits her time away from the sport, in rehabilitation and with her family, for a renewed passion for her trade.

“I couldn’t walk, I couldn’t do all the things that I wanted to do,” Stephens said. “But I did get to hang out with my family and see my little cousin’s soccer games and go to weddings and baby showers and stuff. I loved my time off but when I got back to playing tennis, it was like: ‘This is where I want to be. This is what I love doing. This is fun, this is great, actually.’”

The early returns for Sloane 2.0 have been promising and the best may be yet to come. There are Thursday’s star-spangled semi-finals, only the sixth All-American last four at a grand slam in the Open era and the first since 1981. Then, potentially, an opportunity to become the first American outside the Williams family to win a grand slam singles title since Andy Roddick in 2003 – and the first woman since Jennifer Capriati one year earlier. Not bad for playing with house money.

“I didn’t have any expectations,” Stephens said. “I just was coming out here to play and see what could happen. And I have played. This is what happened.”