High Sierra Topix

After looking over the kids calenders next weekend is our only chance to get out as a family until July. We thought it might be a good time to hit Yosemite before the summer mad season but when I checked out the reservations site there was only one spot available and it's too small for our little pop-up.
I haven't tried to show up without a reservation since we started having kids, anyone know the odds of finding a site without a reservation this time of year? Should we look for a different destination? We haven't been there in years and I swore I'd never go car camping there in summer again....but I'd like for the kids to see the sequoias.

Some very good friends of mine work as assitant hosts in North Pines during the summer and they always seem to have some opened spots available for those without resevations. But, YMMV

Update.

We found some reservations that came free at the last minute. Good thing as none of the FCFS campgrounds are open yet.
Valley was a zoo - but what do you expect. Waterfalls were spectacular.

Last I checked Tuolumne Meadows has half the sites as first come. We had good success heading to Tuolumne and camping for a couple nights, then calling the reservation number and getting sites in the valley for 2 nights- though we had to move between nights one and two.

True enough, but none of the first come/first serve campgrounds open until June. Had I not noticed that 3 days before our trip we would have been in a bit of trouble. Luckily I did realize that and even luckier I picked up some reservations that had been canceled at the last minute. Good time was had by all - however, it will probably be another 10 years before I camp in the valley again.