After being turned down by County Executive Mark C. Poloncarz, leaders of the Buffalo & Erie County Public Library took their plea for an additional $3.3 million to county legislators during a budget hearing Thursday.

There already is a $300,000 increase in library funding in the proposed county budget for 2013, and Library Director Mary Jean Jakubowski said $3.3 million more is needed to restore hours and programs that were trimmed during the last two years.

“While we sincerely appreciate the proposed $300,000 increase in funding, it does not come anywhere near to what the system has requested, nor does it allow us to fully restore open hours, services, materials or staffing levels to support such operations,” Jakubowski told lawmakers.

Poloncarz has proposed a budget that would give the system’s 37 libraries $22.2 million next year, an increase of $300,000. But that fell well short of a public campaign the library conducted to press for $3.6 million more from the county next year to restore library hours and other programs that were cut under then-County Executive Chris Collins.

The library had encouraged patrons to send thousands of postcards to the county executive as he prepared his budget proposal.

“I must stress very clearly that our staff is doing a tremendous job stretching and eking and pulling out every last dime that they can and utilizing any types of economies of scale,” Jakubowski said. “That’s what puts us at this critical juncture. We are walking a very thin line toward having to reduce services and hours further.”

Library officials had hoped to use the funds to keep branches open more hours and to revive library programs that had been reduced when the library’s budget was cut in 2011.

“I don’t think you’re going to find this administration has the same mentality toward the library system as the former administration did,” Deputy Budget Director Timothy C. Callan said. “Given the fiscal realities and problems facing the county for next year, we could not give the library $3.6 million in additional support.”

Poloncarz has proposed a $1.38 billion budget that would increase property taxes by 3.4 percent to fill a budget hole that remained after his staff used surplus funds to balance the budget. The county, he said, is faced with several factors, including declining revenue, increasing costs for health care and other benefits, and dwindling grant money.

It was not clear Thursday how legislators will react to the request from the library as they finalize the budget during the next few weeks. Several legislators have said they hope to eliminate the proposed property tax hike and are looking for areas of the budget to cut.

Jakubowski said after the meeting that she owed it to patrons to make the funding request to restore library services. “Very clearly, we appreciate the situation and the circumstances,” Jakubowski said. “But we are here for the public, and the public is asking us for more services, so we must ask for this additional funding.”