The deaths were reported in the small island province of Biliran, a day after the storm pounded the east of the archipelago nation.
Kai-Tak tore across the major islands of Samar and Leyte on Saturday, toppling power lines in 39 towns or cities and damaging roads and bridges, the national disaster agency said.
Some 87,700 people were forced from their homes in the region. But the previous death toll had stood at just three.
"There is a total of 26 people dead from landslides in four towns of Biliran. We have recovered the bodies," Sofronio Dacillo, provincial disaster risk reduction and management officer, told AFP.
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Kai-Tak weakened on Sunday afternoon, with gusts of up to 80 kilometres (50 miles) an hour, and was reclassified as a tropical depression, state weather forecasters said.
But disaster officials warned that more floods and landslides were possible and said 15,500 passengers were stranded because ferry services remained suspended in parts of the region.
The Christmas holidays are a busy travel season in the mainly Catholic Philippines, with people heading home to the provinces.
The nation is battered by about 20 major storms each year.
Samar and Leyte bore the brunt in 2013 of Super Typhoon Haiyan which left more than 7,350 people dead or missing.
In the Leyte city of Tacloban, Saturday's storm brought flash floods of up to 1.5 metres (five feet) and strong winds that left the city without power and water, according to its disaster office chief.
Bernadas said 82 per cent of Tacloban's districts were flooded.
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