Eleven arrived on the clay of Roland Garros, but only three are left with Bernard Tomic, the only survivor of five Australians involved on Monday.
Hunter won her place by knocking out another Aussie, teen prodigy Tahlia Kokkinis, while Schoolkate defeated Belgian Kimmer Coppejans in a three-setter.
Priscilla Hon and Maddison Inglis were both seeded but lost in straight sets. James McCabe lost to Zdenek Kolar 6-3 7-6 (7-1).
Inglis was beaten 6-1 7-5 by American Robin Montgomery while Hon went down to France's Jessika Ponchet 7-5 6-2.
That left Hunter and 17-year-old wildcard Kokkinis to fight to be the last Australian women standing. Hunter,14 years her senior, won eight of the last nine games to take the tie 6-3 6-1.
Hunter, seeking to make the singles at Roland Garros for the first time since 2023, now plays China's Xiyu Wang. The 30th seed is ranked 45 places above her at 148.
Late on Tuesday Schoolkate joined her in the next stage after defeating Coppejans7-5 1-6 6-4.
He now plays Darwin Blanch, a 229-ranked American teenager.
Tomic, 33, faces Italian teen Federico Cina, ranked 216 in the second round.
Elsewhere Grigor Dimitrov, who had Jannik Sinner on the ropes at Wimbledon last year before his pectoral muscle gave way, lost to Jaime Faria 3-6 7-5 7-6 (10-6).
More successful was Bianca Andreescu, the 2019 US Open champion.
The Canadian has not gone beyond the fourth round at a grand slam since that New York success, during which time, plagued by injuries and a loss of form, she crashed from fourth in the world to outside the top 200.
A pair of wins on the second tier ITF circuit have lifted her mood and her ranking (to 137), and she breezed past French wildcard Daphnee Mpetshi Perricard 6-3 6-2.
Mpetshi Perricard is the 17-year-old sister of men's world No.32 Giovanni and will be better for the experience though she took defeat hard.
Karolina Pliskova, the former world No.1 who made her comeback this year at the age of 34, beat Yeonwoo Ku 0-6 6-3 6-1.
Thirteen Australians are already in a main draw place, 11 through their ranking plus Adam Walton and Emerson Jones via wildcards.
Some of those are warming up elsewhere in the region and had mixed fortunes.
In the Geneva Open Alexei Popyrin eased by Clement Tabur 7-6 (7-2), 6-7 (5-7), 6-4.
In Strasbourg Daria Kasatkina maintained her recent good form, moving into the quarter-finals with a strong 6-1 6-3 win over Peyton Stearns. She'll now meet fourth-seeded Dane Clara Tauson or Romania's Jaqueline Christian.
But in Morocco Ajla Tomljanovic went out to Burundi's 231-ranked Sada Nahimana 6-3 7-5.
AUSTRALIANS ALREADY IN THE FRENCH OPEN MAIN DRAW:
Men: Alex de Minaur (seed), Alexei Popyrin, James Duckworth, Thanasi Kokkinakis, Aleksandar Vukic, Rinky Hijikata, Adam Walton (wildcard).
Women: Maya Joint, Talia Gibson, Daria Kasatkina, Kim Birrell, Ajla Tomljanovic, Emerson Jones (wildcard).
AUSTRALIANS STILL IN QUALIFYING:
Men: Bernard Tomic,
Women: Hunter Storm, Tristan Schoolkate
