There's no set screw on the 2003 Rangers (too simple of an attachment method apparently). Look at where it mounts to the windshield from inside the truck. The center section of that tab is a spring steel clamp that grips the tab adhered to the windshield (you'll be able to see where the clamp is separated from the rest of the mount by the lines showing the edges of the clamp). You'll see a slot on the top and bottom of the mount where it attaches to the windshield. You have to stick a small flat blade screwdriver in between the mounting tab that is adhered to the windshield (again, not the windshield itself) and the mirror mount and apply pressure. Not too much, but enough to release sufficient pressure from the spring steel clamp to allow you to wiggle it off. I went through trying to find this information myself a few weeks ago when I installed an auto-dim mirror. There is no definitive information out there. One guy will tell you one thing, and the next one that comes along is also an authority on it. Others said don't even try it, that it can't be removed w/o breaking the windshield. I'd say they have also changed methods throughout the years. I found it easier to remove it by pulling it down toward the dash as opposed to up toward the headliner.
1) Put the screwdriver in the little slot you see (either on the top or bottom, your choice). Apply twisting and outward (away from the windshield) pressure.
2) While applying the twisting and outward pressure w/ the screwdriver (away from the windshield), wiggle and apply downward (or upward depending) pressure holding onto the arm of the mirror with your other hand.
3) When you see it start to move you'll feel relief and figure out that you'll get it off w/o breaking the freaking windshield out of the truck.
Make sure you're screwdriver is on the mounting tab and not between the windshield and mounting tab. Be careful. I read about several broken windshields by ham-fisted Ranger owners when trying to find the same info. Start off with very little pressure on the screwdriver and arm and proceed from there. You can always increase pressure if a little bit doesn't work, but if you break the windshield, it's over.
Good luck and let us know how it goes.