ALBANY — State Senate Democrats have made it a tradition to pass election-related bills on their first work day of each year, and this year that tradition included an effort to get New York more attention in presidential races.

On Monday, as lawmakers started their first full week back in the state Capitol for the year, Senate Democrats moved to advance a package of election-related bills that they said will help to defend New Yorkers’ right to vote.

Perhaps the most impactful bill in the package would move New York’s presidential primary from the current June date to the first Tuesday of March, aligning the state primary schedule with at least 17 other states and territories that hold their presidential primaries on that date. New York hosts all other primaries for local and statewide office in June, but this law would only impact the presidential primaries for Democrats and Republicans.

“This proposal is really all about relevance,” said Sen. James Skoufis, D-Middletown, the bill’s author and lead sponsor. “As it stands now, presidential candidates only come to New York for one reason, if we’re speaking frankly. And that is, they collect big checks from corporate boardrooms and penthouses in Manhattan, and they fly out that night to some battleground state or some state that is relevant in the primary nominating calendar.”

The political math is basic here: Skoufis and the bill’s backers, mostly Democrats, contend that the earlier a state casts its primary votes for the two major parties in the presidential race, the more relevant those states are to the race, and states that vote later are less relevant.

Presidential candidates demonstrably spend more time in states that vote on Super Tuesday, which often is the day when major party nominees finally cross the threshold to become their party’s candidate for the general election. States that vote later do get less attention, and with New York being a safe state for Democrats, very little attention is generally paid to New York’s presidential primary races.

“What this bill looks to do is make sure that contenders actually come here for other reasons (than fundraising), that they come here to talk to our voters, they come here to elevate our issues,” Skoufis said.

New York once cast its primary votes on Super Tuesday; in 2008 the state’s Republican and Democratic voters cast their ballots on that year’s Super Tuesday election date in February.

The Senate package passed by the chamber on Monday includes one bill that would expressly prohibit “deceptive practices” and voter suppression in elections, and boost the linked penalty for violating someone’s right to vote. Another would bar foreign-influenced businesses from spending money in state and local elections. A third would prohibit dissemination of personal information of election officers, including poll workers and board of elections staff.

A fourth bill in the package would permit any voter in New York to register to vote at a secondary residence with which they plan on maintaining a continuous connection and where they plan to stay connected. That would permit people who own second homes or otherwise split their time to choose to register at an address not otherwise considered their primary residence, but it would not permit people to register to vote at two separate residences.

Another bill in the package will allow the state Board of Elections to create a uniform training curriculum for all state election commissioners, replacing the current system that requires counties to develop a county-specific training program.

All six bills passed the Senate on Monday by a largely partisan margin, with most Republicans opposing most of the bills.

The legislation did not address a major concern among voting rights advocates this year, revolving around a change made at the U.S. Postal Service. No longer will the mail system postmark mail on the date it is received, but instead on the date it is first processed at a mail sorting center.

Especially for rural and sparsely populated regions, this change could delay the postmarking of ballots cast by mail by multiple days, potentially pushing mailed in ballots past the legal deadlines for them to be counted in primary or general elections. New York’s laws require mail-in ballots to be postmarked by the date of the relevant election.

Senate Elections Committee Chair Kristen Gonzalez, D-Brooklyn, has a bill that would address some concerns related to changes in the mail system, but that wasn’t included in this bill package.

“We’re certainly going to push that in this legislative session,” she said.

Gonzalez also batted away criticism from some Republican members of her committee, who have complained for all eight years that the Democratic majority has passed voting legislation in its first day that they’re doing so outside of regular order. Sen. Mark Walczyk, R-Sackets Harbor, has complained about this in prior years. These bills were not allowed to age for three days, as most legislation in Albany is required to do. Instead, lawmakers in the Senate fast-tracked them to allow them to be voted on during the second day of session, the first day of real legislative work.

“That is normal, and a conventional process for the Senate, and it’s a tradition that we’ve had for many years,” she said. “I don’t see it as out of the ordinary.”

This package passed in the Senate as separate bills on Monday. Companion bills in the Assembly have not yet moved for the year, and the Senate does not always move in lockstep with the lower house on legislation.