I will most likely will never thru, but if I did I would definitely include the approach trail. I have walked the approach trail many times and I still get that tingle when I reach the first white blaze, and being that I live right on the trail I see white blazes nearly everyday.
A most excellent plan.
If you don't want to do the approach trail and you don't want to back track there is another option to the top of Springer. Drive another 1.6 miles past the FS42 Springer parking lot to Big Stamp Gap where the BMT crosses FS42. Take the 1.5 mile BMT hike to the top of Springer.