Honestly - Thursday, before 3 if you can swing it. Fridays can be pretty tough, and you might as well get the extra day out of it rather than leaving early enough Friday day. Also I'd plan to try one of the other campgrounds close by if Big Meadows doesn't work out.
I would reserve in advance. Any weekdays in the AM is good except Friday to arrive without a reservation .
I have gone there early on a Friday to find it was too late for a good weather weekend. Big Meadows fills up first but Loft, Lewis and Matthews Arm can fill up quickly as well. Some people put their names on a list for cancellations to open up but you have to wait right there at the camp office. I am single so I bring light gear and can hike into the backcountry when the campsites are full. It is free but not many families can handle backcountry.
As early as possible I would assume. We arrived around 4:45 with our reservation and everything was full.
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