The tiny Bridgehampton school district Tuesday won what many staffers had called a political gamble when voters approved a $12.3 million budget carrying the year's biggest percentage tax hike of any system in Nassau or Suffolk counties.

Sayville and West Babylon, which offered revised budgets with much smaller tax increases, both won large majorities in the revote.

Bridgehampton's budget -- with its 8.76 percent hike in tax collections for the 2014-15 school year -- pulled 240 "yes" votes to 145 "no" votes, garnering approval of 62.3 percent of those voting. The district serves 169 students on the Island's semirural South Fork.

"It's certainly a relief -- momentarily," said Ron White, the board president. He added the district would strive to build public support by making residents more aware of its extensive academic and extracurricular offerings.

The district's state-mandated cap was 4.42 percent, so the spending plan had to pull at least 60 percent approval to override that limit. Bridgehampton's plan got only 54 percent in the May 20 vote.

"They got more people out this time," said one parent, Kelley Davis, 34, as she emerged from the polling site at Bridgehampton School. She added that she and other parents received automated phone calls from the district urging them to vote.

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Bridgehampton school supporters said they want to provide an enriched educational experience for their students, many of whom are sons and daughters of workers in the area's services industry.

Official state figures put Bridgehampton's spending at about $58,000 per student. Per-pupil costs in Bridgehampton are high because of the district's small enrollment.

Before the approval of Bridgehampton's budget, North Babylon was the only Long Island district to succeed in passing its original cap-busting budget on a second attempt since the state's tax-cap limits were imposed in 2012.

Voters in the West Babylon district passed their $99.3 million budget by a wide margin -- 1,432 to 542, or 72.5 percent. Sayville passed its $90 million spending plan by an even larger majority -- 2,518 to 797, or 76 percent.

School boards in both districts lowered their budgets to meet district caps after defeats on May 20.

Sayville's trimmed budget nudges spending up 0.66 percent and has a tax-levy increase that equals the state-imposed cap of 1.22 percent.

Many local voters who supported the spending plan Tuesday said they were swayed by cost-cutting that included elimination of secondary summer school and 15 coaching positions.

"I think they needed -- if this is the right phrase -- to be taught a lesson," said Townsend Anschutz, 69, a safety engineer at Brookhaven National Laboratory.

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West Babylon's budget will raise spending 0.63 percent and taxes 1.36 percent. In trimming that budget, the district cut the equivalent of 9.9 teachers, 18 hall monitors and a number of off-site sports.

Lisa Pastore, 49, of West Babylon said she was against the initial budget and voted "no" last month. Tuesday, she voted in favor of the revised budget.

"The first time I wasn't thrilled," Pastore said. "But they redid it and now I am OK with it. They lowered it a little, but something is better than nothing. . . . The first time, they wanted too much."

With Joie Tyrrell