Your best overall strategy for trail safety would be to seek human walking partners, perhaps through a walking group or social or neighborhood organization. Not only are threats (coyote and otherwise) going to be more hesitant to attack a pack of humans or even a dog restrained in the midst of one, should there be any sort of mishap, humans are more capable of coming to each other's defense, using mobile phones, blowing whistles, rendering first aid or administering medications, etc.
If you do decide to bring a service dog into an area where dogs are prohibited, it would be wise to carefully weigh the reasons for that prohibition, and take steps to maximize the degree to which the dog is serving and not being illicitly taken for a walk.
Something like a visual assist dog would be in effect physically attached to their human via something that is more an arm-extending "harness" than a "leash" - they are not free to sniff around and start at squirrels. In contrast, a dog which serves in another capacity, being walked on a leash has a much higher chance of coming into conflict, in both practice and perception, with the reasons and regulations prohibiting dogs - things like harassing wildlife, the hazard a leash presents to cyclists and even other walkers, fecal contamination in sensitive watersheds, etc. Even if the dog's legal status is sufficient to avoid a citation many of the practical concerns remain, and need to be given real consideration before deciding to proceed.
So if the dog is legally brought into the otherwise prohibited area (preferably in addition to human companions), it would be good to keep the dog on a very short leash or harness so that she is constrained in both fact and appearance to remain by your side for the safety and consideration of all, including herself. If possible a backpack carrier would be even better. Or if the path is paved and not hilly, a jogging stroller might work especially if enclosed. Remember also to bring water and provide it at frequent breaks - dogs cannot match the sun and heat tolerance of humans.