United Airlines Holdings Inc. said it would give staff who have so far declined to be vaccinated against Covid-19 an extra five weeks to get inoculated as the Sept. 27 deadline approaches for the company’s mandate.

Chicago-based United was the first U.S. carrier to mandate staff vaccination and on Wednesday said it would put employees exempted on medical or religious grounds on unpaid leave and would look for ways to later reintegrate them through enhanced testing and masking.

Airlines have taken different approaches to employee vaccination. They have tinkered with incentives such as bonuses and extra time off. More recently, several carriers have announced testing protocols for unvaccinated workers and other penalties for those refusing to get the shots.

United announced the most stringent policy. The company said last month that it would terminate employees who aren’t vaccinated by the airline’s deadline unless they had an approved religious or medical exemption.

United has declined to specify how many staff remain unvaccinated, but the number has more than halved since the airline announced the mandate, which the company said had also had a positive impact on hiring.

United said its plans were complex, underscoring the difficulty companies in many sectors have in developing plans to enforce mandates.

Delta Air Lines Inc. stopped short of mandating Covid-19 vaccines, but will require unvaccinated employees to pay an extra $200 a month for health insurance to help cover additional costs, such as hospitalization. The airline will also require regular testing for employees who haven’t been vaccinated.

Several carriers, including Delta, American Airlines Group Inc. and Alaska Air Group Inc. have said they would stop paying for work that unvaccinated employees miss due to Covid-19 infection or exposure. That means unvaccinated employees who have Covid-19 or have to quarantine would need to dip into their own banks of sick time or take unpaid time off.

Alaska is also requiring newly hired employees to be vaccinated—a step that Delta and United have also taken.

In mandating vaccines, United hasn’t been followed by its domestic and international partners, including regional airlines that fly a large portion of its passengers. SkyWest Inc. and Mesa Air Group Inc. fly hundreds of daily flights on behalf of United and encourage employee vaccination, but have stopped short of a mandate.

Few overseas carriers have introduced mandates, including United’s partners in the global Star marketing alliance of airlines that sell each others’ flights.

Australia’s Qantas Airways Ltd. , a member of the rival Oneworld airline grouping that includes American, last month said it would require front-line workers including cabin crew, pilots and airport workers to be fully vaccinated by Nov. 15. Remaining employees must be vaccinated by March 31. Qantas said an employee survey showed 4% of staff were unwilling or unable to be vaccinated.

Write to Doug Cameron at doug.cameron@wsj.com