HONG KONG—Apple Inc. is adding new contract manufacturing partners in Asia to boost production of its smartphones and tablets, people familiar with the supply chain said, as the technology company tries to ensure adequate supply to meet consumer demand.

Later this year, Apple will outsource manufacturing of the cheaper version of its latest iPhone, the iPhone 5C, to Wistron Corp., a Taiwan-based contract maker that currently produces smartphones for BlackBerry Ltd. and Nokia Corp., the people said. Apple also plans to add...