Probably the "best" driving route in late June is via Tioga Pass. It's the shortest and most scenic. Cautions? Follow the speed limit. Best place for an overnight stop? Maybe June Lakes, Mammoth Lakes, Bishop ... depends on your travel style.
Where are you staying in Yosemite and for how many days? Will you be spending your exit day enjoying Olmstead Point, Tenaya Lake, and Tuolumne Meadows? Are you detouring to Bodie State Park? Do you need to arrive in Death Valley early in the morning on your arrival day, or will you be spending time enjoying the drive?
Here is a past thread with information from various TA members about that drive, tho' they were talking about a September drive which is much warmer in Death Valley: tripadvisor.com/ShowTopic-g61000-i315-k35083…
This thread also has good info, tho' the Topic is misleading: tripadvisor.com/ShowTopic-g28926-i29-k368014…
another vote for tioga pass its probably the most spectacular drive iv ever had. within an hour at one stage you can go between heavy snow and freezing temperatures to the death valley floor and over 120 degrees. the scenery is just incredible
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