Democratic presidential hopeful Massachusetts' Senator Elizabeth Warren speaks to supporters at a rally outside the New Hampshire State House, after signing papers to officially enter the New Hampshire Primary race in Concord, New Hampshire on November 13, 2019.

Just as the holiday shopping season kicks into overdrive, presidential candidate Sen. Elizabeth Warren on Tuesday introduced a measure to protect part-time workers, which companies from Target to UPS hire to help handle extra work during the peak period for retailers.

The measure would require large employers to offer employees more hours before hiring new employees or subcontractors. It would also allow part-time workers to participate in an employers' pension plan and be eligible for family and medical leave.

The bill, which was also introduced by Rep. Jan Schakowsky, D-Ill., is the latest example of Warren's push to introduce legislation that protects worker rights as the income gap has widened and record corporate profits have made lower-income workers feel left behind.

Warren is among the leading Democratic candidates for president, and she is jockeying for the support of labor groups and unions with the likes of fellow progressive Sen. Bernie Sanders and front-runner Joe Biden, the former vice president.

Unions, which typically favor Democratic candidates, remain influential in rallying voters. Amid a crowded Democratic field, the AFL-CIO — one of the largest federations of U.S. unions — has said it is withholding its presidential candidate endorsement until February. That's when the first four nominating contests of the primary election season occur — in Iowa, New Hampshire, South Carolina and Nevada.

Warren framed the proposed "Part-Time Worker Bill of Rights Act" as a matter of fairness and economic security.

"For far too long, companies trying to boost their profits have taken advantage of part-time workers by assigning them unpredictable work schedules — creating real hardships for them," said Warren, of Massachusetts. "My legislation with Congresswoman Schakowsky puts an end to this practice by giving part-time workers the rights, stability, and other protections they deserve to build better financial futures for themselves and for their families."

Schakowsky accused companies of using part-time status to "rig the system and maximize profits while exacerbating income inequality."

Staffing for retailers can be particularly hard to predict because elements such as a snowstorm can throw a dent in traffic, and thus the need for staff.

Retailers have experimented with software systems to allow for elements such as weather and to let workers sign up for work on a case-by-case basis, said Craig Rowley, a senior client partner at executive recruiting firm Korn Ferry. Such programs would give workers the ability to pick and choose the hours they want to work and give retailers a better shot at ensuring their workers will show up. That technology, though, is still in its infancy.

"The industry has addressed these issues in a lot of ways already, but they still have the issue of there's a retail store in the northeast and a snowstorm, so you really didn't need as many people on staff," he said.

Many of the recent improvements for part-time workers have come as a tight labor market has forced retailers to offer better benefits in a fight for human capital. It's also led many to rely more heavily on current workers. A survey done by Korn Ferry found that 63% of respondents are also planning to give permanent workers more hours this year when they aren't able to find people to fill shorter-term roles. Walmart said in 2017 it would give its employees extra hours during the holiday season, rather than offering those hours to seasonal workers.

Target has said it is offering its more than 130,000 seasonal workers a minimum wage of $13 an hour, in line with its starting minimum hourly wage for full-time workers. A spokesperson for the retailer said that in preparation for the holiday season, "we always start by having conversations with our current store team members to understand their availability and interest in working additional hours."

UPS is paying its 100,000 seasonal workers $14 per hour and offering tuition reimbursement for those who are eligible. Part-time employees who work for the company on a permanent basis are eligible for healthcare benefits. They may also be eligible for pension, depending on job and location.

— CNBC's Lauren Thomas and Jacob Pramuk contributed to this report.