If it's Mt Graham two things--
Start early, the bottom will be a rotissere
and it is a little tricky to discover the proper dirt road/Jeep trail that leads to the lower TH.
IMO better to go down, then up again, at least you'll be ascending into cooler climate. I camped below Shingle Mill trail junction and saw no one else that evening or the next morning the whole way out as I climbed up.
I was there in August of last year, got rained on , and had plenty of water in the stream.
Anywhere below falls is likely to be low trafficed. May have to look for water as I am not sure how well the stream is going, falls photos I have seen looks like very low this year so far.
If it's the Galiuros--way too hot, even higher up as Bassett Peak only a little over 7000 feet.