You NEVER need to start a new character. Your pugilist can actually change to ANY other class. Jobs are basically advanced versions of classes that can be unlocked at level 30 for their class, and basically "layer" on top of the class (they share exp, among other things), with the exception of the three Heavensward classes, which need you to be level 50 and up to a certain point in the main storyline (when you enter Ishgard), and they start at level 30 instead of level 1.