Since you are still planning and doing section hikes, I might suggest you plan your hikes South to North unless you like up hill hiking.
Reason, Big Meadow to Mathews Arm is all up and down and not critical, but if you hike North during the last 10 miles toward Front Royal you be going more down hill and with the flow of other hikers.
Your timing seems right - it gets too hot for me starting in June.
Also, go with the weather forecast more than a calendar date. No need to mention bear safety - just hang your food items.
Thanks Oldpappy, I was actually starting to think that going South to North might be a good idea. I just have to figure out the logistics of getting to and from wherever it is I decide to set off from. And I am definitely planning to start based on the weather. Fortunately I have some flexibility with my schedule at the moment as far as being able to take time off.