On its face, it was Sheriff Marlin Gusman on the wrong end of Tuesday’s ballot. He had asked voters to let him redirect an existing property tax earmarked for construction projects, hoping to use the money instead for operating costs at Orleans Parish Prison, and the voters said no.

But in truth, the rejection of the ballot measure represents a setback for Mayor Mitch Landrieu. The Sheriff’s Office may be in charge of running the facility, but City Hall is responsible for funding it. So it is the mayor who will have to come up with any extra money Gusman needs to make jail improvements spelled out in a federal court order.

The bright spot for Landrieu and the city’s budget on Tuesday came in the form of strong voter support for a ballot measure that may clear the way for higher property taxes to bolster police and fire protection, two of the city’s biggest costs.

The City Council will still have to put those tax hikes in front of voters again. Tuesday’s vote was a statewide referendum that simply lifted the cap on how much Orleans residents can decide to tax themselves: 10 mills instead of 5 mills for each of two departments, Police and Fire. But the apparent support — 59 percent of the vote in New Orleans — for even a potential tax hike can only be encouraging for city officials trying to balance a precarious budget.

“There’s a cloud over the administration’s head: finding creative ways to fund those consent decrees,” said local pollster Silas Lee, referring to court orders demanding reforms at both the jail and the Police Department.

Lee speculated that voters simply had a better grasp on what they were being asked to approve with the police and fire taxes. The sheriff’s ballot measure was part of a broader funding plan with several moving parts, and there was little active campaigning to get it passed. “Something like that, you really have to explain to people,” Lee said.

That could indeed be the broader takeaway for other public bodies hoping to ask voters for extra funding in the near future.

The Audubon Nature Institute tried and failed to win a new long-term property tax for upkeep of the zoo, the aquarium and other attractions earlier this year. Critics blasted the proposal in part because they felt the institute hadn’t really explained why it needed the money. (There also was little urgency for it, given that the institute’s existing tax doesn’t expire for years.)

Next month, the Orleans Parish School Board — which itself is divided over the idea — will be asking voters to approve a ballot measure that is just as knotty and complicated as the sheriff’s. The board wants to redirect money that is being used to pay off old bonds, gradually shifting the dollars into separate accounts for upkeep of the city’s new and renovated school buildings.

Eventually, other city agencies, including the library system and City Park, may also be lining up to ask voters for more funding.

It isn’t clear yet what the direct fallout will be from the “no” vote on Gusman’s tax proposal, which would have provided about $8 million a year in operating revenue for the jail. No one knows exactly how much funding Gusman will need in any given year to satisfy a federal judge that he is making progress toward better conditions at the notoriously troubled lockup.

Estimates have ranged from $10 million to $22 million over and above what the city was paying on an annual basis before the consent decree went into effect. And that could mean a total of as much as $45 million or so a year. The mayor’s budget proposal for 2015 includes $28.6 million for the sheriff, a $4.4 million increase from 2014.

The final sum could fluctuate based on orders from U.S. District Judge Lance Africk, who is overseeing the consent decree, and the vagaries of negotiations between the Sheriff’s Office and third-party vendors brought in to provide various services at the jail.

For instance, this summer, Gusman awarded a $15 million-a-year contract to a Tennessee company that has taken over providing medical and mental-health services, though it’s not clear whether the Sheriff’s Office will reap a savings from no longer having to provide those services in-house.

A “yes” vote from local voters on extra money for police and fire protection would certainly ease the budget situation, though not until 2016. The earliest the City Council could schedule a local referendum on those taxes would be sometime next year, long after property tax bills for 2015 will have gone out to residents.

Perhaps a best-case scenario for the city would be a repeat of what happened in 2014: an unexpectedly healthy increase in overall tax collections. With new retail outlets like Costco and Wal-Mart opening their doors, city officials kept boosting their sales tax estimates as the year went on. Another year like that would give the city some much-needed breathing room.