In the more recent Star Wars: Rebels, it shows Leia was working as a planner and operative several years before the Battle of Yavin. While under her father's cover as a member or aide for the Imperial Senate, Leia was planning operations to aid resistance cells at age 16. The plan to get her own ships stolen so that Alderaan could remain blameless was hers. The inspiration to get the plan going again even when things seemed hopeless was hers.
After the loss of Alderaan, Leia likely became the leader of Bail's group of Rebel cells. The war was now fully on and she had no planet to be a politician over and the senate was disbanded. She was the leader of the former Alderaan-supplied Rebel cells now. The only person she would actually need to answer to is Mon Mothma. General Rieekan is in charge of the base. Leia is in charge of the unit, so she'd be the regional general, while Rieekan is the base commander.
However, Princess Leia is one of those that has to be doing something. So while Rieekan is in charge of the base and is leading the defense of said base, Leia is in charge of the evacuation of said base. She's the one giving orders to the fighter pilots that will be escorting the transports off Hoth. She's overseeing the evacuation and gives the order to send two transports at a time. When Han comes in after the Command Center has been hit, She's dispatching orders for troops to help cover the remaining evacuation forces now that the shield generator is gone. Her last order is to sound the final retreat and evacuate their remaining forces. That is where she is in command. The base has fallen, so her duty is to evacuate the remains of her regional cell command, troops, and as much equipment as they can.
We don't see her doing this later, because she's cut off from her cell by being on the Falcon.
The rescue operation for Han Solo is a core cell operation. The Heroes of Yavin are going after one of their own while keeping the risk of other Alliance cells to a minimum. They could have attempted a military operation, but that would have drawn Imperial attention even on Tattooine. Having a Jedi helps with these sorts of operations, based on what they use to do before the Clone Wars.
With the Alliance gathered, Leia's command defaults to Mon Mothma as overall commander-in-chief. The operation had been planned already, and Leia's command skills were unneeded in a fleet action, nor did she have a base of operations that was vital to the operation. What the Alliance needed was ships and Starfighters for the Death Star, and an elite team for the ground mission. So she was able to volunteer for General Solo's shuttle command team. Leia had long ago proven she's a good shot with a blaster and adapts fairly well to the situation. Also pretty much unbreakable to torture if captured. Also the Heroes of Yavin have worked as a good team for several years at this point. If nothing else, that would be a morale booster for the Alliance ranks to know they are the ones on the job of taking down the shield around the Death Star II.