Hulk in early June?

Hello MP peeps.  Planning first trip to the Sierra in many, many years.  While the approach will obviously be very snowy this year is early June a reasonable time temperature-wise to be climbing classic alpine rock routes on the Hulk and so forth?   Maybe a dumb question but I really don't know!  Thanks.

Temperatures should be somewhere between fine and bearable, but the snow on the approach will in all likelihood be pretty grueling... There are a few steep talus fields which would suck pretty hard if they are covered in compacted neve-like snow/ice. Might want to bring some combination of crampons/ice axe in case those sections aren't fully melted.

Semi-related: see Chris Mac's trip report of a winter ascent here: http://www.supertopo.com/tr/Incredible-Hulk-Winter-Ascent-A-First/t12927n.htm

We're from the PNW.  Steep snow is not a problem! Thanks.