Trim tough stems from remaining chard; separate leaves from ribs. Tear leaves into large pieces; chop ribs into bite-size pieces. Heat remaining 2 tablespoons oil in a large skillet over medium. Cook onion, stirring often, until starting to soften, about 5 minutes. Add chard stems, season with salt and pepper, and cook, stirring occasionally, until crisp-tender and onion is soft, 5 to 8 minutes.