If the paved road at the top of the ridge (El Camino Cielo?) is still open, you can park at Cold Springs Flat trailhead and hike 5 miles (one-way) to Little Caliente. That's how I always go, because I wouldn't want to risk driving my little Toyota Corolla over that treacherous dirt road, even when the gates are open. It's a nice hike. You do have to cross the Santa Ynez river, but my guess is that's likely wadable. There's some occasional-mucky bushwacking near Gibraltar reservoir, but it's doable.
Big Caliente is also doable from the same parking area, but it's a *much* longer hike (9 miles, one-way). I've done it as a day-hike, but you need to get started early and move quickly. If you're looking to camp, you can be more relaxed about it. In fact, I would almost recommend Big Caliente precisely *because* the road to it is closed right now. My understanding is that it's usually the more visited of the two springs in the area, so the non-ease of access right now might make it less rowdy and therefore more enjoyable.
Pix of Little Caliente hike from there here.
Pix of Big Caliente hike from there here.
The last pix in both albums have topo maps, and links to a precise hiking description (for Little Caliente), and links to what I've drawn out on Googlemaps.
Joel