Hey Domer,
Long story short, you might need to get used to driving if you want cooler hikes in the Tucson area. Most of the hikes that start at lower elevation trailheads in the Catalinas, Rincons, and Ritas will still be hot and nasty for a couple of miles until they achieve some elevation and\or overhead cover. One option is to start out nice and early and get to 4000+ feet before 8 or 9 AM, to beat the heat, so-to-speak. The second option involves driving, specifically to higher elevation trailheads. If you want to stay in the pines and out of the heat, you'll need trails that loop or gain elevation, as we have a lot of trailheads in the mountains here that are the "ending point" for hikes starting out in the foothills, and they'll just take you back down into the heat.
First thing to do is go to Summit Hut (or wherever) and purchase these four maps:
"Rainbow Expeditions: Southern Arizona Hiking Club Maps - Santa Catalinas"
"Rainbow Expeditions: Southern Arizona Hiking Club Maps - Chiricahuas"
"Rainbow Expeditions: Southern Arizona Hiking Club Maps - Santa Ritas"
"Rainbow Expeditions: Southern Arizona Hiking Club Maps - Rincons"
They'll set you back about 25 bucks for the set. They are basically compositions of various topo maps with established trail systems and known points of interest highlighted. If you're going to be in the Old Pueblo for a while then having these sitting around will save a lot of time and confusion.
On the Catalinas map (or on the great one Joe built on the HAZ,
here), take a look at the network of trails near the summit of Mt. Lemmon, specifically trails like: "Wilderness of Rock", "Marshall Gulch", "Aspen"; and also those that start near Palisades and circle north and west like "Butterfly Peak", etc. Basically look for trails or sets of connections you can make with series of trails that start on the north or west side of the highway and don't loose too much elevation. These trails will start in pine and basically stay there. Don't bother with anything lower than Bear Canyon, at first. The only lower elevation trail I could recommend in the heat is Ventana. If you start out early enough, it's not bad. The canyon limits exposure quite a bit.
The Rincons and Santa Ritas are trickier. In the Santa Ritas, many of the trails that start out or quickly head toward the south side of the range are more temperate. Basically the trails like "Chinaman" and "Cave Canyon" are cooler, temperature-wise, as are most in Gardner Canyon. Be forewarned that this is a multi-use area. Some of the trails leading into Gardner are actually FSR's and used to be frequented by ATV traffic, but that may have changed. The Rincons won't be your friend on this issue, I'm afraid. All of the approaches to Wrightson from the traditional "front" (west) slowly climb up from the valley floor. On the "backside" is Turkey Creek, which starts out low and quickly climbs the east side of the Rincons. Miller Creek is another option, too. Both will start out with a trek through the desert but quickly get to Chaparral.
The Chiricahuas are another story, altogether. It's a drive, but it's well worth it. There are dozens of trails that never leave pine and take you past awesome rock formations of all descriptions. Heart of Rock is as good a place to start as any, so is Echo Canyon.
Other places to investigate that are off those maps include the Pinaleño Mountains and the Galiuros. Most of the trails that start near, or lead into, Mt. Graham are pine trails. The Nuttal fire closed a couple of good ones in that area, but many are still open for business. Also check out the Dragoons. Middlemarch hikes through piñon pine and never gets too hot (That I remember). In the Quinlans (Kitt Peak) and further West there are a few lesser-known hikes that are usually pretty temperate and offer awesome wildflower views. Anyway, hopefully that will give you somewhere to start...
Short of driving, your only other option is to do like nutty natives like myself and embrace the searing sun and savage, relentless heat as you fling yourself ever higher into the foothills, craving the reward that awaits you when you achieve higher, cooler elevations. It's one way to beat the crowds!
Jeff