Don't overlook the Vivian Creek Trail on San Gorgonio. Very steep (the first section of the trail) and a butt-kicker for a day hike. I actually hiked this trail about a month after I went up Whitney for the first time (first week in June - just like you!) and was impressed with the difficulty.
One important tip: reduce your training down to almost nothing for a few days before your hike to let your body recover from the intense training.
The north chutes are a better place to train for the most difficult part of an early June MMWT trip, the chute. We ended up using crampons and axes at about 10,350' in late April. I'm sure there is a route to the summit beaten in along with high speed glissade runs.
Also, the approach is about 6 miles to these chutes, similar to what you will have getting to Trail Camp; however, it is best to go cross country once you reach S.F. Meadow.
