The MMWT is unique in the Eastern Sierra in that it is the only trail which require a permit for day hiking, if you want to go beyond Lone Pine Lake.
My experience is the chance of you not getting a permit cancellation are low. If you do not get one, there are so many places to go in the immediate area you will not leave disappointed.
My personal likes in the area are the Meysan Lake Trail and Cottonwood Pass/Trail Pass Loop. Meysan Lake Trailhead is within Whitney Portal and end at Meysan Lake some 5 miles away. I love the Cottonwood Pass/Trail Pass Loop with a stop on the 11,500' Trail Peak. This peak offers some of the best views in the Sierra.
