The Dungeon Master's Guide describes how one can modify existing races or create new (playable) ones. Pp. 285-287, specifically.
However, there are more official playable races than just those in the Player's Handbook. You may want to take a look at What are the playable D&D races in 5e, and where can I find them?
More broadly, the DMG is largely designed around the concept that D&D should not only allow you to create characters and adventures, but also spells, settings, races, new classes... your own game. I view it as a worked example of how to create an RPG in the D&D style. If you're looking for more than the Player's Handbook provides, the DMG likely gives some advice on how to get there.