John Muir Wilderness is home to some great Alpine lakes and scenery. A good number of entries into the John Muir Trail start at John Muir Wilderness. There are some great day hikes (albeit a bit long) to experience John Muir Wilderness, one of the really good ones is Little Lakes Valley that leads you to Gem Lake.
https://www.alltrails.com/trail/us/california/little-lakes-valley-to-gem-lakes
The trailhead is about 45 minutes from Mammoth Lakes. It is a popular hike and the trailhead gets full early in the day during weekends, during peak season. The overflow lot is a half mile away and that gets full soon as well, so get there early.
Within a mile of moderate climbing, you get to see this gorgeous view of Little Lakes Valley.
Along the trail, there are plenty of “little” lakes to stop by. Heart lake and Box lake are very accessible from the trail.
After about 5 miles of hiking, you get to Gem lake and clear Turquoise waters. Even in early September, this lake was cold and woke you up.
Gem lakes area is popular for short backpacking trips. Backcountrycow has a great overview of a typical backpacking trip. https://www.backcountrycow.com/blog/backpacking-little-lakes-valley-to-gem-lakes
I highly recommend this as either a day hike or a short backpacking trip in John Muir Wilderness.