june will be better for the east side sierra nevada (horseshoe meadows, whitney portal, big pine creek, onion valley, hwy 168/bristlecone forest, south lake, rock creek, tioga pass) but too late for death valley (too hot even at night).
east side sierras dictate sunrise starts because of the high desert influence/heat and the looong, hard climbs.

if you must do may, west side sierras are the ticket. ton of big climbs with fewer snow concerns. climbs generally not as steep as east side sierras but longer. pick your poison.

may works for tucson and mt. lemmon, mt. graham and kitt peak (altho kitt peak was closed as of this last february) as long as they are all sunrise starts.

for all of these areas, would pack a windbreaker, skull cap, arm warmers, long-fingered gloves and possibly a neck gaiter for mornings till 9:30am-10am. obvs a pita but comfort matters and one or more items
may come back into play for descents.

should you decide to hit southern california instead of central california/sierras instead, let me know. plenty of hc climbs from san diego area north to santa barbara area w/o snow, goathead and (potentially) traffic concerns.
big mountain forays in may/june in socal call for sunrise starts but w/o the extra gloves, gaiter and skull cap.

and frankly, unless you've got a bee in your bonnet about a particular climb for whatever reason, the mount lemmon climb from tucson is more scenic than any of the east side sierra climbs save ca hwy 120/tioga pass. on the west side
sierras, personally recommend ca hwy 198 (generals hwy) southern entrance into sequoia national park.