MANILA—Typhoon Rammasun veered slightly south early Wednesday, sparing the Philippine capital from the full brunt of the storm, whose howling winds, heavy rains and storm surges have left a trail of destruction in many provinces since it made landfall at dusk Tuesday.

The weather bureau said Rammasun's eye will pass over Tagaytay, a mountain resort city just south of metropolitan Manila, before heading to Bataan province around noon Wednesday and then out to the South China Sea, where it is expected to intensify before moving...