I somehow had the impression you were going to Show Low on business, and wanted to get in a hike in addition to that. If you had said you were going to Show Low just to hike I would have responded differently. Like, why Show Low?
If you plan a trip to the White Mtns. be sure to also check the local newspaper's website for hike write-ups. Their web address is wmicentral. I had forgotten to mention that earlier. But truthfully, the best hikes in the White Mtns. are in summer, and they would be up around Greer, Big Lake, Black River, Hannagan Meadow, etc. There is some nice hiking around Pinetop, also. Heber has some interesting areas, such as Chevelon Canyon, but the ranger station there is not very helpful in helping you find the hikes. I called them once to ask about Chevelon, and they knew nothing, and then I found out that a lot of other people also called them about it, and learned nothing. They couldn't even tell me if there would be water in the creek there! They referred me to someone who is part of the USFS fire unit there. I asked him if there would be water in the creek because I was taking a bunch of friends there for a two-night backpack. He said, "Yeah, there might be a few pools along there, you might find a little water." When we got there and found that there is a ton of water in there and that it is one swimming hole after another for 8 miles between the Crossing and the Dam, I realized he was trying to keep me OUT of that place, by making it sound like there wasn't much water. Not a very good public servant! You will have better luck with the Lakeside and Springerville ranger districts.
Alpine RD, on the other hand, you have to go there and get the maps, because the people who answer the phone in that office know absolutely NOTHING. They are total idiots. I tried to find out the length of the Fish Creek trail and they tried to tell me it was 6 miles, when I could see on my maps it had to be 12. Turns out it is 12.
Also, over by Springerville the Game and Fish dept has developed three wildlife areas with hiking trails. They are really neat to see. There is Wenima, White Mtns. Grasslands and Sipe Wildlife Area. Since I live by Springerville I am familiar with trails and other places to hike over by there. To me it's preferable to hikes around Pinetop because I rarely see anyone, even on big holiday weekends, if I choose my hikes well. The main problem we have up there right now, though, is trying to protect our designated non-motorized trails agains ATVers.