Hi Several options , first if you keep checking Rec.gov space will open up and you can grab a day/overnight permit.
Next watch the weather when we start to get storms late June early July that last for a few days or a week, many will not show up, Storms are a few hours a day. If you are experienced with hiking not a problem. Easy to get a walk in permit from the Visitor Center 1 mile south of Lone Pine for the next day.
Next option is wait till Mid September Monday -Thursday and most days you can get a day permit for the next day.
October bring a friend there might be very few people on the mountain ,watch for early storms that may dump 2-5 feet of snow at the upper elevations, drifts that cover the trail from just above Outpost Camp area.
No permits at the Trailhead Make sure you stop in town and get your permit ALL PEOPLE THAT HAVE THE LETTER!!!!!! you need to confirm and pick your permit up at the Visitor Center 1 Mile South of Lone Pine Very limited Hours so plan ahead don't wait they close around 4 :30 PM