Tuckahoe’s school board voted Tuesday night to attempt an override of the state’s tax cap for a second time — an effort that will require a 60 percent majority if the tiny East End district is to avoid a tax freeze next year.

Tuckahoe’s three trustees decided unanimously to submit a $19,293,000 budget to local voters on June 21, the date set statewide for school-budget revotes. The proposed budget represents about a 1 percent spending increase, but less than the district sought originally.

Under that budget, Tuckahoe’s tax levy — total revenues raised through property taxation — would rise 1.75 percent. Superintendent Allan Gerstenlauer said the increase would add $81 to the average homeowner’s annual tax bill.

Trustees voiced optimism over the budget’s chances of passing, based on pledges of support from about three dozen parents and others who attended an earlier board meeting on May 23. Still, there were warnings of the dangers of low voter turnout.

“If all the people who were at the meeting last week go out and talk to their neighbors, the budget will pass,” said board vice chairman Robert Grisnik, a local pharmacist who has served on the panel for more than 30 years. “If they don’t do due diligence, then we don’t know what will happen.”

Tuckahoe’s initial $19,533,000 budget proposal for 2016-17 was one of only two district spending plans among Long Island’s 124 school systems that failed in balloting on May 17.

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The vote was 146-145 — a bare majority, far short of the 60 percent required to go above the district’s state-imposed tax cap.

Elwood, the other district on the Island where a proposed budget was rejected, already has decided to keep within its cap when it holds a revote. June 21 is the date set by the state for any districts scheduling second attempts.

Under state law, any district losing a second override attempt would be hit with a tax freeze next year. Residents would lose state tax-rebate checks of $130 due for distribution in the fall.

Tuckahoe is among the 10 smallest districts on the Island, with 350 students in its single school building, which houses classes from prekindergarten through eighth grade. An additional 135 students are sent to high school in Southampton on a tuition basis.

Like other East End school systems, Tuckahoe has experienced dramatic demographic changes in the past 10 years. The percentage of Latino students in the district has doubled from 30 percent to 60 percent, and 20 percent of students speak limited English.

Tuckahoe school supporters cited services, including all-day prekindergarten and elementary class sizes averaging 17 students, as evidence that the district provides strong support for enrollees, including those who need extra help.

Beth McNeill-Muhs, the mother of a kindergartner, said she particularly likes school-sponsored “mindfulness” workshops for students and parents, which emphasize the need for awareness of other people’s situations and feelings.

“That tells me our school is really dedicated to finding new tools that enhance our children’s educations,” she said before Tuesday night’s meeting.

Gerstenlauer, who serves as Tuckahoe’s part-time interim superintendent, said the district will save $244,000 because voters on May 17 approved a five-year tuition contract for its high school students with the Southampton district. Tuckahoe formerly contracted with Southampton on a year-to-year basis and will get a discount under the extended agreement. The tax levy rise of 1.75 percent is below the 2.48 percent proposal rejected by voters last month but well above the state’s 0.38 percent cap.

East End districts have managed in past years to pierce caps at somewhat higher rates than districts farther west, largely because their overall taxes are lower. Bridgehampton won a revote override in 2014, with a 62.3 percent majority.

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Some Tuckahoe residents said that particular circumstances within their district could make it more difficult to muster the required 60 percent voter supermajority. One lingering issue is the debate over the merger of the district with Southampton.

In 2014, Southampton residents voted down a merger plan, with 972 in favor and 1,066 opposed — the second defeat in just over a year. Tuckahoe residents strongly supported dissolving their district into Southampton’s, 533-25.