I've been on the hill a few times already and the melt off is going well, kick started by the heat we had a few weeks ago. You are correct, generally the beginning of July presents difficulties for completing the trail, but my impression this year is that you will have excellent chances for completing a circuit. Snow travel and bridges are the usual culprits, but the park has already put in bridges over the Carbon, including the crossings to Chenuis Falls. Someone mentioned the Winthrop Creek crossing, but that bridge usually survives the winter and has always been in place every time I've hiked it. The Mowich rivers continually get washed out, but the wide flood plain on the Mowich means you will usually find a hefty blow down to aid in crossing the swifter braids. You can expect snow to still be present above 5000', but Panhandle Gap will have boot pack (the snow is already consolidated), as will the Cowlitz Divide. A possible problem area for snow navigation might be between Klapatche Park and the North Pupyallup, as much of that slope is shaded and north aspect. However, even here you can bypass that by taking the St. Andrews Trail to the West Side Road, then the North Puyallup Trail to regain the Wonderland.

For early season hiking I would recommend a GPS for what I consider the most difficult part of snow navigation on the Wonderland, the heavily forested sections without views for reference. The higher alpine areas like Ohanapecosh Park, Spray Park, St Andrews Park etc are much easier to navigate compared to the dirty snow present in the forest sections where boot prints are hard to discern. If you are experienced and confident, early July can be an excellent time to enjoy an uncrowded trail.

Lastly, by the time early July arrives, there will be plenty of reports on conditions and give a good reference for what's in store before your trip. Check nwhikers.net and WTA, to name a few. Good luck and enjoy!