If you want to combine this with the Narrows, access can be spotty in May and sometimes early June because of high water flows. It all depends on the snow melt that year. 2010 and 2011 were big snow years and water was too high most of May and into June. They close the Narrows at around 130 cfs.
May is generally pretty warm/hot in the Grand Canyon. It's just a fact of life that there can be a 50-60 degree swing between night-time temps at the rim and daytime temps at the river. So, if you're camping on the rim as well as at the river, you either have to contend with freezing temps at night on the rim *or* sweltering heat at the river. Nearly impossible to avoid both.
For warm hikes below the rim and generally low flows in the Narrows, you can't beat October, imo.
For outside of the park boundaries (and no permit issues), check out Kanab creek wilderness (north rim side.) By May, trailheads should be accessible though sometimes they're not...
Personally, I'd go north, not south.