In terms of section hiking, the GET should be doable westbound starting late May at ABQ, which is around the time eastbound thru-hikers finish up there. Heat and sun will be issues, certainly, but quite possibly manageable for a trail-hardened AZT thru-hiker. Ending at Glenwood NM, ie avoiding Arizona at that time of year, keeps you above 4800 feet throughout the hike, and often above 6000' (New Mexico is far and away the better of the two states to hike during June, at least given this route's layout.)
Water may become a concern depending on how this winter shapes up, as conditions are currently very dry.
Also, FYI, an updated mapset to the route will be available in time for spring hiking.
GET in the Blue Range (from photo album)