The Sierra Club is such an organization. Besides doing environmental advocacy on a nationwide level, they also have local chapters where people organize activities like hikes. As the name suggests, they started in California and are most active there, but there are chapters throughout the country.
Around Big Sur, it looks like the Ventana and Santa Lucia chapters would be the ones to look at. Each one has a schedule of upcoming hikes and other "outings"; you'll see they are pretty frequent and range from easy to challenging.
Each outing has contact info for a leader, so when you find a hike that interests you, you'll need to contact the leader to reserve a space (most hikes are limited to a fixed number of hikers so that the group does not become excessively large). The leader will also give you info on where and when the group will meet (typically they meet at some central location and carpool) as well as anything else you need to know. You should mention that you're not a club member, but normally this should not be a problem, especially since you're visiting. If they suggest that you join the club, usually the membership dues are not very expensive.
My experience has been that Sierra Club hikes put substantial emphasis on safety and preparedness. You'll want to make sure you have appropriate hiking gear, perhaps following the ten essentials system. As a traveler, you may not be able to carry all of this on a plane with you, so you might have to plan a shopping trip. The trip leader may be able to offer advice. Also, you'll likely be asked to sign a liability waiver at the beginning of the hike, so that you don't sue the Sierra Club if anything bad happens.
(Disclosure: I used to be a member of the San Diego chapter until I moved away.)