If you have a choice I suggest Sept. As WB suggested snow and road closures will likely limit your choices in May.
I second Steve's suggestion of 1000 lakes. Also, you can continue on past Shadow Lake to Lake Ediza (and beyond) if you like.
Then there is always the hike to the top of Mammoth Mountain. You can either go from the Mammoth Lodge side, or from Twin Lakes up the "Dragons Back).
Mammoth also offers many trailheads within a 10 to 15 minute drive. Crystal Lake, Mammoth Crest, Duck Lake/Pass, etc...... Many options to turnaround early, on continue on for longer hikes.
If you do not mind a 45 minute drive you can drive to the trailhead at Saddlebag Lake for more options. Also, the turn off to Saddlebag lake is very near the Tioga Pass entrance to Yosemite, so many many more hikes to choose from.
Tarn below Mt. Conness
lastly, Gaylor Peak (near Tioga Pass Entrance to Yosemite Park)
I really like this area due to the wide selection of one day hikes (or shorter) within short driving distance.