Dog-friendly parks similar to Arboretum?

I recently adopted a senior dog only to discover that she has a paralyzing fear of cars/traffic. I plan on finding a dog behaviorist at some point to see if that issue can be resolved but in the meantime, she needs her exercise and going for walks in my neighborhood just isn't feasible right now.

I took her to the Arboretum the other day and she LOVED it. I think it's far enough away from any cars and traffic noises that she can relax and just enjoy the new smells and being outdoors. I definitely plan on making Arboretum trips a more common occurrence each week but I was looking for suggestions on other places in town that allow dogs and might have a similar feel to them so that we can get some variety. We tried both the 3-mile loop at Memorial Park and the trails at Terry Hershey park; unfortunately, those are pretty close to major roads and don't feel quite as "wild" so she got fearful at both places.

We live in the Upper Kirby area but I'm willing to drive a bit especially on weekend mornings when we have more time. Anyone know of similar places that are dog-friendly and have a lot of tree cover, quiet trails, distance from traffic, etc.? TIA!