There are mining roads in every direction from Silverton that can get you into areas where you can "dispersed camp," as San Juan National Forest calls it. One of the most scenic is in the flats of the Ironton area just north of Red Mountain Pass.
Your truck may be a little big for most of the jeep roads in the Silverton area. But, while you are there, you might try two of them on for size. One is Engineer Pass, which is up above Animas Forks and continues over to Lake City. The other is Ophir Pass, which goes from US 550 NW of Silverton to Ophir and Telluride. I have been all over the San Juans on jeep roads and would recommend staying off most of the other passes with a truck the size of yours. Too many 10-point turns on the hairpins with a big truck in places where my Ford Ranger and Nissan Frontier only had to do 4-point turns!
Incidentally, if you are going to go over to Telluride while you are there (very highly recommended for the scenery), there is a USFS campground with 19 electric hookups and 10 pull-thru sites. It is located on CO 145 south of Lizard Head Pass and is called the Cayton campground. And while there, you can try trout fishing in Trout Lake just north of the pass and, better yet, Alta Lakes above Ophir.
Finally, if you are going to visit Mesa Verde NP, try the Juniper Campground at McPhee Reservoir. It has paved sites with electrical hookups, flush toilets, showers, and even a little shade in spots. Much nicer, less crowded, and cheaper than Morefield Campground inside the national park. And, while there, you can stop right next door and visit my favorite Anasazi museum anywhere in the Four Corners region.