MANILA—Abu Sayyaf, once written off as one of the global jihadist movement’s also-rans, is gaining strength in the southern Philippines by chasing down high-value victims at sea and ransoming them off for millions of dollars.

After a relative lull for most of a decade, kidnappings have surged to more than 20 annually since 2014, when the group’s main leader Isnilon Hapilon swore allegiance to Islamic State.

That...