I need to take off my shower head for cleaning and possible replacement.
I can see there is Teflon tape on the fitting where the head connects to the arm. When I reconnect the head, can I reuse the existing tape? Do I wrap an extra layer around the existing tape? Or do I need to go ahead and completely remove and replace the existing tape?
You should remove the tape and replace it, a small roll at a big box store is less than a dollar.
I agree with Ed Beal that you should replace it, but if you're in a pinch you certainly can reuse PTFE tape, assuming you recover enough of it and it's not totally shredded.
Carefully unwrap it and evaluate which end is in better shape. Re-wrap with that portion to the outside, respecting thread direction (you want the tape to tighten as you thread the parts together, not loosen or fold back). Usually that's clockwise looking from the pipe end.
Check carefully for leaks as the seal will not be as reliable.
As an additional answer alluded to by Nelson, it also somewhat depends on the use case. If it's a shower head (as in this example) where a drip really won't make any difference, if you don't have new, go ahead and reuse. However, if it's hidden or difficult to get to, you'll want to do it right the first time. Also, if your jurisdiction permits the use of Teflon tape on natural gas or other flammable gas/fluid connections, always use new, as a leak there could be catastrophic.
For an application like this where a slow leak would not be a disaster you can omit the tape or substitute something similar like cling-wrap.
That's if the tape is even needed, usually shower-heads use a washer.
Teflon tape is a one-time use item. It's sole purpose is to deform and tear itself into the threading of the pipe to prevent water from getting through the threads.
Clean the old tape off as much as possible. Use a metal wire brush to really get off the crusty stuff as well but don't scratch up the chrome/nickel/whatever finish and apply about 2-3 layers of new tape clockwise; assuming that righty-tighty applies here. This should take no more than 5 minutes.
Do note that if the shower head has a rubber gasket then that gasket is what produces the water tight seal and teflon tape may not do anything if the gasket fails. It all depends on how that joint was designed. In this scenario you can still apply the teflon tape as long as you don't put any tape where the rubber gasket meets the pipe.
