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In the second case, you could also drive the Apache Trail after Tonto. I think however you may not have as much time as you think. It also sounds like this is soon. If so, keep an eye on the weather. There are possibly some major storms coming which would affect most of the higher areas of AZ in the northern and eastern parts of the state. This could include the drive from GCNP to Payson.
Beautiful drive from Camp Verde via Strawberry and Pine to Payson and before Payson you'll find Tonto Natural Bridge. Just a short side trip and worth it imo. We drove the unpaved part of Apache Trail several times with rental cars and had no problems. Just drive slowly and not when wet. Lake Roosevelt has a nice visitor center you can visit before the drive on AT or on the way to Show Low if you drive that way.
Thanks Tet and Bob for the info. I think we're going to skip Tonto National Monument this time. We're going to have to come back, anyway, now that I look around southern Arizona more. We've always done Sedona, GC and Utah. I'm going to post our upcoming itinerary in a separate post. Hope you'll feel free to comment. Always appreciate all the help on TA. Every trip we plan using TA exclusively, and they're always once-in-a-lifetime experiences! Bless you all!