I'm afraid that to get to the cooler temperatures, you probably have to drive a little bit farther, making a return to the valley by 1pm a bit more difficult. But if you are hiking early on a Saturday morning, some of the lower elevation spots may still be tolerable, especially if near water. You might want to check out the Horton Creek Trail near Payson (
http://hikearizona.com/decoder.php?ZTN=62). This one might kick your butt a little bit as it climbs steadily the whole way, but it also parallels a beautiful creek. It'll get a little bit warmer by afternoon, but a summer morning should be nice. It's a popular spot, and you'll encounter other people and dogs, so if yours is off-lead be prepared for what you might encounter.
The trails of the Cabin Loop (
http://hikearizona.com/decoder.php?ZTN=797) are probably a great spot, though as the summer dries out, there are sections where there is no water in the creeks, but it's at 7500 feet so you and your dog should have no problem making it from one creek to the other. This area takes almost 3 hours to get to however, which limits the amount of hiking you can do in the morning (or how early you get up) to be back in Glendale by 1. It is definitely not ideal for your schedule. The entire loop itself is far too long, but you can hike much smaller portions of it with ease as it crosses forest roads in many places. I would recommend the northern half of the Houston Brothers trail as a starter. Beginning at the Pinchot Cabin, hike south to at least Aspen Springs (about 1.5 miles) or McFarland Spring (2 mi.) and then return to your car, which would double the mileage. There's plenty of dispersed camping available within a short drive of the trailhead (camping is not allowed at the trailhead).
Another spot up north that might be really nice is the Bear Canyon Lake trail. (
http://hikearizona.com/decoder.php?ZTN=1210) This will also be a tough one to get back by 1pm, but plenty of dispersed camping. The trail on the western side of the lake is quite easy and well-traveled. On the eastern shore, it's less of a trail and I would probably recommend making this an out-and-back rather than trying to circle the entire lake.
I'm not too familiar with the spots that are more accessible from the west valley, so perhaps somebody will point you to some better stuff along the I-17 corridor that's a bit shorter of a drive for you to get to.