Wow! I belive the Diamond O is right along the Evergreen road into Hetchhetchy! Striking distance to a relatively underused treasure trove of early season Sierra Pleasure. One of these is:
The Lake Vernon Loop Trek.
I can suggest that you pehaps stay at Diamond O the first night, after driving. The next day you can get a wilderness permit at the Hetchhetchy entrance station about a mile beyond Camp Mather. For a three day early season hike in early June.. I recommend the Lake Vernon Loop. You can cross the O' Shaunessey Dam(through a several hundred foot long tunnel in solid granite) and climb the switchbacks toward Laurel Lake, perhaps even stay at Laurel the first night(Good Fishing even early May but beware the crossing of Frog creek can be treacherous).
By the way.. When you get to Beehive Meadow you will be within a few feet of "The Gem" of Northern Yosemite. The spring of which I speak is located at the end of the meadow near the junction of the Laurel lake/Lake Vernon trails. It is a small wooden box.. and despite it's appearance, it contains the absolutely sparkling clear ice cold "nectar of the gods" spring water of The Beehive Spring.
And having just climbed 3000 feet in elevation over 6 miles you will no doubt appreciate this small miracle!
Proceed to Lake Vernon(fantastic sub-alpine scenery just 10 miles from the dam) and camp there.
The granite flats below Lake Vernon make for incredible adventuring and in early June you should be able to cross the lake Vernon bridge of falls creek easily (though some years water is high and crossing might mean wet feet) Check the water level when you get there. I have found it has a reverse tide, being lower in the afternoon than morning.
The beauty of this hike is that even if you can't or don't want to cross falls creek there is enough to do at Vernon to make a very respectable 3 day trip out of it all if you decide to then just hike back down to the Dam. The trip from the Dam to lake Vernon alone requires NO creeks be crossed though some minor patches of snow might exist upon approaching the Moraine ridge junction and later along the shoulder of Mt Gibson(If you choose to do the loop).
Falls creek is SPECTACULAR as it thunders down canyon from this point! Fishing at Vernon can be good some years but never as good as at Laurel at the same time of year. Best to fish at Laurel and explore at Vernon.
Proceeding on the trail up over Mt Gibson the views of The Grand Canyon Of The Tuolumne River explode into view. There could be a minor amount of snow as you reach the highest elevation of the trip along the shoulder of Mount Gibson.
Next You drop down into Tiltill valley. (Bring Bug Dope for this section!) The meadow in Tiltill is fantastic in early june! Tiltill Valley is a strangely contorted Granite canyon and a little known but worthwhile lunch stop.
Next you decend to Rancheria falls which is offers much flat granite sun basking and fishing opportunities in Hetchyhetchy itself.
The 6 mile hike out takes you under Wapama and in sight of Tueeulala Falls. Be aware that the road into and out of Hetchhetchy is locked after 9PM and opened by 7AM
Also there is a very nice(but HOT) backpackers campground ($5/night 1 night upon entering or exiting only) at the dam itself with bathrooms and running water, and even a wash sink.
Steel bearboxes are numerous there as well.
My introduction to backpacking and the High Sierra began with this hike! I cannot recommend this area highly enough. Particularly if you want to remain far from the crowds of "The Valley".
The Hetchhetchy area could be just the ticket.
Cheers, Hetchy