Yes. "In earnest" has a wider meaning, "seriously" or "for real." "She picked up her pace and was soon running in earnest" means that she started by walking faster, but not really running. Then she was soon "running in earnest," not just walking fast, not just jogging, not just trotting, but really running. Running seriously, SINCERELY believing that she was in danger and SINCERELY trying to escape.