* S.Korea reports 1,212 new daily cases

* Movement restrictions extended in Seoul

* Officials warn tougher curbs could be reinstated

* Surge being fuelled by young and unvaccinated (Adds report of Wednesday cases, Seoul mayor comment, number of variant cases)

By Sangmi Cha

SEOUL, July 7 (Reuters) - South Korea reported its second highest number of daily new COVID-19 cases on Wednesday, just days after it began easing social distancing restrictions in some parts of the country, buoyed by an accelerated vaccine rollout.

With the majority of the 1,212 new cases as of midnight Tuesday coming from densely populated Seoul, officials extended movement curbs in the capital and surrounding regions for at least another week and are considering pushing restrictions back up to the highest level.

According to Yonhap news agency, Wednesday's cases were also expected to top 1,000, with 1,010 already tallied by 6 p.m. (0900 GMT)

Prime Minister Kim Boo-kyum said a fourth wave of the virus was spreading rapidly, especially among unvaccinated people in their 20s and 30s, while a growing number of highly contagious Delta variant cases raised new worries.

Kim urged people in that demographic to get tested preemptively "to protect not just yourself, but everyone in your family, friends, school and the country."

"If the situation is not under control after monitoring for two to three days, it might leave us with no choice but to impose the strictest of all social distancing levels," Kim said.

President Moon Jae-in ordered the military be mobilized to aid wider contact tracing and urged authorities to open additional testing centres in densely populated areas, presidential spokeswoman Park Kyung-mee told reporters on Wednesday.

The daily caseload was the highest since Dec. 25, when South Korea was battling a third wave of the pandemic.

Officials had been moving in recent weeks toward a full reopening of the country. Movement restrictions in much of the country were eased on July 1, although officials in greater Seoul held off as they watched case numbers beginning to creep up again.

Health experts said the relaxation of measures that restricted business operating hours and social gatherings outside of Seoul, along with the knowledge that further easings would be coming, led to public complacency, particularly in socially mobile younger people in the capital.

Seoul Mayor Oh Se-hoon called on the prime minister to consider expanding vaccinations to younger people, which he said would alleviate the situation.

Around 85% of the new locally transmitted cases were in the Seoul metropolitan area, which is home to more than half of the country's population.