What are Hestia's spheres of influence?

Question

For my cleric character, she worships Hestia. I want to know what domains Hestia has control over(besides Life) so I can better determine how my cleric should progress in her faith.

Answer 1

Her sole domain is Life

That's all the rulebooks ascribe to her. See below for more details.

Hestia is not a 'core' deity for any of the D&D settings

Hestia is the IRL Greek goddess of hearth and home. D&D 5E's Player Handbook includes 4 RL pantheons as alternative options for players to use. The extent of what we have on her for 5E is this:

The Celtic, Egyptian, Greek, and Norse pantheons are fantasy interpretations of historical religions from our world’s ancient times. They include deities that are most appropriate for use in a D&D game, divorced from their historical context in the real world and united into pantheons that serve the needs of the game.

and for the Greek Pantheon specifically

The gods of Olympus make themselves known with the gentle lap of waves against the shores and the crash of the thunder among the cloud-enshrouded peaks. The thick boar-infested woods and the sere, olive-covered hillsides hold evidence of their passing. Every aspect of nature echoes with their presence, and they’ve made a place for themselves inside the human heart, too.

and that she's a Neutral Good deity of the Life domain, whose symbol is the Hearth.

That's it. That's all there is about her in D&D because she's not part of any of D&D's invented settings. She's a real world mythological figure.

Thus, to answer your question simply: she has no other domain but Life. Because D&D's domains are a D&D invented concept, and they've only assigned Life to Hestia.

So, most importantly...

Talk to your DM

If your DM has decided that a Greek goddess is permitted in your campaign, you need to talk to your DM about how they fit into their setting.

Because (from the same Appendix A linked above):

Your DM determines which gods, if any, are worshiped in his or her campaign. From among the gods available, you can choose a single deity for your character to serve, worship, or pay lip service to. Or you can pick a few that your character prays to most often. Or just make a mental note of the gods who are revered in your DM’s campaign so you can invoke their names when appropriate. If you’re playing a cleric or a character with the Acolyte background, decide which god your deity serves or served, and consider the deity’s suggested domains when selecting your character’s domain.

Outside of that, you're going to need to look at other sources outside of D&D, because Hestia is an import. If you want to be historical about it, well, here's her wikipedia article.