The Cimarron River is pretty darned good for fishing; people stay in the surrounding Cimarron Canyon State Park. There are a couple of campgrounds there.
Just to the north in the Valle Vidal. It's a fairly primitive area with a gravel road passing through. Fishing is excellent. Folks pull in RVs to camp.
North northwest of Taos is the BLM Wild Rivers Rec Area with campground, where the gorge carved by the Red River meets the Rio Grande gorge. You have to hike down to the river. Again excellent fishing, but the hike is a good one.
Finally, some folks stay below Taos at the BLM Orilla Verde Rec Area just outside of Pilar. Runs ride beside the Rio Grande. Again, campground is available. The fishing is ok, I've heard.
Finally, if you into lake fishing, do Eagle Nest Lake State Park. I think your rig will fit in there ok.
For the two state parks, I'd recommend reserving if you can. It's popular season.